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MaplePrimes Activity


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Maple can solve the easiest two problems of the Putnam Mathematical Competition 2018.  link

 

 

Problem A1

 

Find all ordered pairs (a,b) of positive integers for which  1/a + 1/b = 3/2018

 

eq:= 1/a + 1/b = 3/2018;

1/a+1/b = 3/2018

(1)

isolve(%);

{a = 1358114, b = 673}

(2)

# Unfortunalely Maple fails to find all the solutions; eq must be simplified first!

(lhs-rhs)(eq);

1/a+1/b-3/2018

(3)

numer(%);

-3*a*b+2018*a+2018*b

(4)

s:=isolve(%);

{a = -678048, b = 672}, {a = 0, b = 0}, {a = 672, b = -678048}, {a = 673, b = 1358114}, {a = 674, b = 340033}, {a = 1009, b = 2018}, {a = 2018, b = 1009}, {a = 340033, b = 674}, {a = 1358114, b = 673}

(5)

remove(u ->(eval(a,u)<=0 or eval(b,u)<=0),[s]);

[{a = 673, b = 1358114}, {a = 674, b = 340033}, {a = 1009, b = 2018}, {a = 2018, b = 1009}, {a = 340033, b = 674}, {a = 1358114, b = 673}]

(6)

# Now it's OK.

 

Problem B1

 

Consider the set of  vectors  P = { < a, b> :  0 ≤ a ≤ 2, 0 ≤ b ≤ 100, a, b in Z}.
Find all v in P such that the set P \ {v} can be partitioned into two sets of equal size and equal sum.

 

n:=100:
P:= [seq(seq([a,b],a=0..2), b=0..n)]:

k:=nops(P): s:=add(P):
numsols:=0:

for i to k do
  v:=P[i]; sv:=s-v;
  if irem(sv[1],2)=1 or irem(sv[2],2)=1 then next fi;
  cond:=simplify(add( x[j]*~P[j],j=1..k))-sv/2;
  try
    sol:=[];
    sol:=Optimization:-Minimize
         (0, {x[i]=0, (cond=~0)[], add(x[i],i=1..k)=(k-1)/2 }, assume=binary);
    catch:
  end try:
  if sol<>[] then numsols:=numsols+1;
     print(v='P'[i], select(j -> (eval(x[j],sol[2])=1), {seq(1..k)})) fi;
od:
'numsols'=numsols;

[1, 0] = P[2], {5, 10, 13, 14, 16, 19, 21, 23, 24, 26, 28, 31, 32, 35, 36, 38, 40, 41, 43, 46, 47, 49, 52, 53, 55, 57, 59, 62, 64, 67, 69, 70, 73, 75, 76, 79, 81, 82, 85, 86, 88, 90, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 106, 107, 109, 112, 113, 115, 118, 120, 121, 126, 128, 135, 138, 144, 150, 154, 159, 165, 168, 170, 171, 172, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 219, 222, 225, 228, 229, 230, 231, 235, 236, 237, 238, 240, 241, 242, 243, 246, 248, 252, 253, 254, 258, 260, 261, 264, 266, 267, 270, 273, 274, 275, 276, 279, 280, 284, 287}

 

[1, 2] = P[8], {3, 7, 10, 13, 14, 16, 19, 21, 22, 25, 27, 28, 31, 32, 34, 36, 37, 40, 42, 44, 46, 48, 50, 52, 55, 58, 59, 61, 62, 64, 66, 69, 70, 73, 74, 76, 77, 81, 83, 85, 87, 88, 91, 93, 94, 97, 98, 100, 102, 104, 106, 108, 110, 112, 115, 117, 118, 121, 122, 124, 126, 132, 135, 139, 140, 141, 144, 151, 153, 159, 171, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 215, 216, 217, 218, 219, 220, 222, 223, 225, 229, 230, 231, 234, 236, 237, 238, 240, 241, 242, 243, 246, 249, 250, 252, 255, 257, 258, 259, 261, 263, 266, 267, 269, 273, 276, 279, 281, 282, 284}

 

[1, 4] = P[14], {7, 10, 13, 15, 19, 21, 22, 24, 26, 28, 31, 32, 33, 37, 38, 40, 42, 44, 47, 48, 50, 52, 55, 56, 59, 60, 63, 64, 66, 68, 71, 72, 75, 76, 78, 81, 82, 87, 88, 90, 91, 94, 95, 96, 97, 100, 102, 103, 105, 106, 109, 110, 113, 114, 115, 116, 117, 118, 121, 122, 124, 127, 132, 135, 138, 142, 144, 145, 154, 156, 159, 160, 163, 165, 170, 171, 172, 176, 177, 178, 179, 180, 181, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 216, 219, 220, 222, 225, 226, 228, 229, 231, 232, 234, 235, 237, 239, 240, 241, 244, 245, 247, 248, 250, 251, 253, 254, 255, 258, 261, 262, 264, 265, 267, 268, 270, 272, 273, 276, 284}

 

[1, 6] = P[20], {10, 14, 15, 16, 19, 22, 24, 26, 28, 30, 31, 34, 35, 37, 40, 43, 44, 46, 47, 51, 52, 54, 56, 58, 60, 62, 65, 66, 68, 70, 73, 75, 76, 79, 82, 83, 84, 85, 86, 87, 88, 90, 93, 94, 96, 97, 99, 102, 103, 106, 107, 109, 110, 114, 116, 117, 120, 121, 123, 126, 127, 128, 129, 130, 135, 138, 142, 144, 147, 152, 153, 154, 156, 157, 162, 163, 172, 174, 177, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 223, 224, 225, 226, 228, 229, 233, 234, 235, 236, 237, 241, 242, 246, 248, 249, 250, 251, 252, 253, 255, 258, 259, 262, 263, 265, 267, 269, 270, 273, 275, 276}

 

[1, 8] = P[26], {4, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 29, 30, 34, 36, 37, 38, 40, 43, 44, 46, 48, 51, 54, 55, 56, 58, 61, 63, 64, 66, 70, 71, 73, 74, 78, 79, 80, 82, 85, 86, 88, 90, 92, 94, 97, 100, 102, 103, 106, 108, 109, 112, 113, 115, 118, 120, 121, 123, 129, 133, 135, 141, 144, 150, 158, 159, 160, 163, 168, 169, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 219, 220, 222, 225, 226, 228, 230, 231, 232, 233, 234, 237, 240, 241, 243, 245, 246, 249, 250, 255, 257, 258, 259, 260, 261, 262, 264, 265, 267, 269, 270, 273, 279, 281, 282}

 

[1, 10] = P[32], {3, 4, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 37, 38, 40, 43, 44, 46, 49, 51, 52, 54, 58, 61, 63, 64, 65, 67, 68, 70, 73, 74, 78, 79, 81, 82, 85, 86, 88, 91, 94, 96, 97, 100, 103, 104, 106, 108, 109, 112, 113, 115, 117, 119, 121, 126, 131, 132, 136, 138, 144, 147, 148, 153, 156, 160, 161, 168, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 219, 220, 222, 224, 225, 226, 227, 228, 231, 233, 234, 236, 240, 242, 243, 245, 246, 248, 249, 251, 252, 254, 255, 257, 258, 261, 262, 264, 266, 267, 269, 270, 273, 275, 276, 279, 282}

 

[1, 12] = P[38], {7, 10, 14, 16, 19, 22, 25, 28, 29, 31, 34, 35, 36, 37, 40, 43, 46, 47, 48, 49, 52, 54, 57, 59, 61, 63, 64, 67, 68, 70, 72, 75, 76, 79, 81, 82, 85, 87, 88, 90, 93, 94, 96, 98, 99, 100, 101, 103, 106, 108, 109, 111, 112, 114, 117, 118, 121, 122, 124, 132, 135, 141, 150, 151, 158, 159, 162, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 219, 221, 222, 223, 224, 225, 228, 229, 230, 231, 232, 234, 235, 237, 239, 240, 243, 244, 246, 249, 250, 251, 252, 255, 258, 259, 260, 261, 264, 267, 270, 273}

 

[1, 14] = P[44], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 49, 51, 52, 54, 57, 58, 61, 62, 64, 65, 69, 71, 72, 73, 76, 78, 80, 81, 84, 85, 88, 90, 93, 94, 95, 100, 102, 103, 106, 109, 111, 112, 113, 115, 117, 120, 121, 124, 127, 132, 133, 138, 141, 145, 150, 153, 159, 162, 165, 168, 171, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 224, 225, 226, 227, 228, 229, 230, 234, 235, 236, 238, 241, 242, 243, 244, 247, 248, 249, 252, 256, 258, 259, 260, 261, 264, 267, 268, 270, 272, 273, 275, 276}

 

[1, 16] = P[50], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 53, 54, 56, 58, 61, 63, 64, 67, 68, 70, 72, 74, 76, 78, 82, 84, 85, 87, 88, 90, 92, 94, 96, 99, 103, 104, 106, 107, 109, 111, 115, 116, 118, 121, 123, 127, 132, 138, 141, 143, 147, 156, 159, 160, 168, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 215, 216, 217, 218, 219, 222, 225, 226, 227, 228, 231, 232, 233, 235, 238, 240, 242, 243, 244, 246, 247, 249, 250, 252, 253, 255, 256, 258, 259, 260, 261, 264, 267, 270, 272, 273, 275, 276, 281}

 

[1, 18] = P[56], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 60, 62, 64, 66, 68, 70, 73, 75, 76, 78, 80, 82, 85, 86, 88, 89, 93, 94, 96, 98, 100, 102, 105, 106, 109, 110, 113, 114, 116, 118, 119, 121, 123, 129, 132, 136, 144, 148, 153, 168, 169, 170, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 217, 218, 219, 221, 222, 223, 225, 229, 230, 234, 235, 237, 239, 240, 243, 244, 245, 246, 249, 250, 251, 252, 254, 255, 256, 257, 258, 259, 260, 261, 263, 264, 266, 267, 269, 270, 273, 276, 279}

 

[1, 20] = P[62], {3, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 63, 64, 67, 68, 72, 73, 74, 76, 79, 80, 81, 82, 85, 87, 88, 90, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 114, 116, 117, 120, 121, 126, 127, 132, 135, 141, 147, 153, 155, 163, 169, 172, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 228, 229, 232, 234, 235, 237, 238, 240, 243, 244, 246, 248, 249, 253, 254, 255, 258, 261, 262, 264, 267, 268, 270, 273, 274, 275, 276, 279, 282}

 

[1, 22] = P[68], {12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 73, 74, 75, 76, 78, 79, 82, 83, 84, 85, 88, 90, 91, 93, 94, 96, 98, 100, 103, 104, 109, 111, 112, 114, 115, 117, 119, 121, 124, 129, 132, 136, 138, 141, 142, 150, 153, 156, 159, 160, 168, 169, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 216, 217, 218, 219, 220, 224, 225, 226, 228, 230, 231, 232, 233, 237, 238, 240, 241, 243, 247, 249, 251, 252, 253, 254, 255, 258, 261, 262, 264, 267, 270, 273, 274, 275, 276, 279, 282}

 

[1, 24] = P[74], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 78, 79, 81, 82, 85, 87, 88, 90, 92, 94, 97, 99, 102, 105, 106, 109, 111, 112, 113, 115, 117, 120, 122, 126, 127, 129, 132, 133, 138, 139, 144, 147, 153, 160, 165, 166, 168, 169, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 225, 227, 228, 229, 230, 231, 232, 233, 234, 237, 239, 240, 244, 245, 246, 247, 249, 251, 252, 253, 255, 258, 261, 262, 264, 267, 270, 273, 275, 276, 277}

 

[1, 26] = P[80], {3, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 82, 83, 84, 86, 88, 91, 92, 95, 96, 99, 100, 103, 104, 106, 108, 110, 112, 115, 117, 120, 121, 124, 125, 126, 127, 128, 130, 132, 138, 139, 141, 145, 150, 153, 159, 162, 166, 174, 178, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 218, 219, 221, 222, 223, 225, 226, 228, 229, 231, 233, 234, 235, 236, 237, 240, 242, 243, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 267, 268, 270, 273, 274, 275, 276, 279, 282}

 

[1, 28] = P[86], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 90, 91, 92, 94, 97, 98, 101, 102, 104, 106, 109, 111, 112, 115, 116, 119, 120, 123, 124, 129, 132, 134, 136, 141, 147, 153, 160, 162, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 221, 224, 226, 227, 228, 233, 234, 236, 237, 240, 241, 243, 244, 245, 246, 247, 249, 250, 251, 252, 254, 255, 257, 258, 259, 260, 261, 263, 264, 266, 267, 270, 273, 276, 279}

 

[1, 30] = P[92], {7, 10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 93, 94, 97, 98, 100, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 119, 120, 121, 122, 123, 126, 127, 132, 135, 141, 147, 153, 157, 165, 171, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 225, 226, 228, 229, 231, 234, 235, 236, 237, 240, 241, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 267, 268, 270, 273, 275, 276, 279, 281}

 

[1, 32] = P[98], {10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 102, 105, 106, 110, 111, 112, 113, 114, 118, 120, 121, 123, 124, 126, 127, 128, 129, 130, 135, 138, 144, 150, 153, 156, 157, 160, 162, 166, 167, 171, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 214, 215, 216, 217, 219, 221, 222, 223, 225, 227, 228, 229, 232, 233, 234, 235, 237, 238, 240, 243, 244, 245, 246, 247, 249, 251, 252, 253, 255, 258, 261, 262, 264, 267, 270, 273, 275, 276, 281}

 

[1, 34] = P[104], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 95, 97, 100, 101, 103, 106, 108, 110, 113, 115, 116, 118, 120, 122, 126, 132, 134, 138, 141, 147, 152, 160, 161, 171, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 219, 220, 222, 223, 225, 226, 228, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 248, 249, 251, 252, 254, 255, 257, 258, 260, 261, 264, 267, 270, 273, 276}

 

[1, 36] = P[110], {10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 117, 118, 120, 121, 124, 125, 129, 134, 135, 136, 138, 139, 141, 142, 144, 145, 147, 150, 151, 154, 156, 162, 171, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 221, 222, 223, 225, 226, 227, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 281}

 

[1, 38] = P[116], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 120, 121, 124, 125, 129, 135, 136, 141, 147, 150, 156, 158, 165, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 256, 258, 261, 262, 264, 267, 268, 270, 273, 274, 275, 276, 279, 282}

 

[1, 40] = P[122], {7, 10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 124, 126, 129, 135, 136, 141, 147, 153, 156, 159, 160, 163, 166, 168, 170, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 242, 243, 244, 245, 246, 247, 249, 251, 252, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 281}

 

[1, 42] = P[128], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 130, 132, 138, 143, 144, 150, 153, 154, 157, 171, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 256, 258, 261, 262, 264, 267, 270, 272, 273, 276, 278, 279, 281, 282}

 

[1, 44] = P[134], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 138, 140, 144, 150, 159, 162, 168, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 249, 250, 251, 252, 253, 255, 256, 258, 261, 262, 264, 267, 268, 270, 273, 275, 276, 281}

 

[1, 46] = P[140], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 159, 162, 168, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 249, 250, 251, 252, 253, 255, 257, 258, 261, 262, 264, 267, 268, 270, 273, 275, 276, 277}

 

[1, 48] = P[146], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 155, 156, 159, 162, 165, 166, 168, 172, 176, 177, 178, 179, 180, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 229, 231, 233, 234, 235, 237, 238, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 267, 269, 270, 273, 275, 276, 281}

 

[1, 50] = P[152], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 156, 159, 160, 162, 171, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 232, 233, 234, 235, 237, 238, 240, 241, 243, 244, 245, 246, 248, 251, 252, 254, 255, 257, 258, 260, 261, 263, 264, 267, 270, 273, 276}

 

[1, 52] = P[158], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 156, 157, 160, 163, 165, 171, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 232, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 281, 282}

 

[1, 54] = P[164], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 156, 157, 160, 162, 163, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 229, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 250, 251, 252, 254, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 281, 282}

 

[1, 56] = P[170], {7, 10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 156, 157, 160, 162, 163, 165, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 255, 258, 261, 264, 266, 267, 270, 273, 275, 276, 279, 281}

 

[1, 58] = P[176], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 156, 157, 160, 162, 163, 165, 171, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 238, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 280, 282}

 

[1, 60] = P[182], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 156, 157, 160, 162, 163, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 238, 240, 243, 244, 245, 246, 247, 249, 251, 252, 255, 258, 261, 264, 267, 268, 270, 273, 276, 279, 282, 285}

 

[1, 62] = P[188], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 156, 168, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 228, 229, 231, 233, 234, 235, 236, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 265, 267, 270, 273, 275, 276, 279, 281}

 

[1, 64] = P[194], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 153, 156, 157, 160, 162, 163, 171, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 225, 226, 228, 229, 231, 233, 234, 235, 237, 240, 242, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 281}

 

[1, 66] = P[200], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 132, 137, 141, 144, 150, 153, 157, 168, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 250, 251, 252, 253, 255, 258, 261, 262, 264, 267, 270, 273, 275, 276, 277, 279, 282}

 

[1, 68] = P[206], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 128, 133, 138, 141, 144, 150, 156, 157, 159, 160, 162, 168, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 221, 222, 223, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 267, 268, 270, 273, 275, 276, 279, 282}

 

[1, 70] = P[212], {7, 10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 126, 132, 135, 138, 141, 145, 150, 153, 157, 159, 161, 165, 166, 168, 171, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 249, 250, 251, 252, 255, 258, 261, 264, 267, 270, 275, 281}

 

[1, 72] = P[218], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 112, 114, 118, 119, 121, 122, 124, 126, 132, 133, 138, 141, 144, 150, 159, 166, 169, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 215, 216, 219, 220, 221, 222, 223, 224, 225, 226, 228, 231, 233, 234, 235, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 259, 260, 261, 263, 264, 266, 267, 269, 270, 272, 273, 276, 279}

 

[1, 74] = P[224], {3, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 101, 103, 105, 107, 109, 113, 114, 116, 117, 121, 122, 124, 126, 129, 135, 141, 143, 147, 152, 153, 160, 165, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 217, 219, 220, 221, 222, 226, 228, 231, 233, 234, 235, 237, 240, 241, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 267, 268, 270, 271, 273, 275, 276, 279, 281, 282}

 

[1, 76] = P[230], {7, 10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 92, 93, 95, 97, 100, 102, 103, 105, 107, 109, 111, 116, 117, 120, 121, 123, 125, 127, 131, 135, 141, 143, 144, 148, 153, 156, 157, 163, 168, 169, 171, 172, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 228, 229, 231, 233, 234, 235, 236, 237, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 264, 267, 270, 273, 275, 276, 290}

 

[1, 78] = P[236], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 88, 89, 91, 93, 95, 97, 100, 103, 104, 107, 108, 110, 113, 114, 116, 118, 120, 122, 126, 128, 134, 135, 139, 141, 147, 150, 153, 156, 159, 162, 165, 167, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 214, 215, 216, 217, 218, 219, 220, 223, 225, 226, 228, 229, 231, 232, 233, 234, 235, 237, 238, 240, 243, 244, 245, 246, 247, 251, 252, 253, 255, 258, 261, 262, 264, 267, 268, 270, 273, 275, 276, 281}

 

[1, 80] = P[242], {7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 75, 79, 81, 82, 84, 85, 87, 90, 91, 96, 97, 99, 100, 102, 104, 107, 109, 110, 112, 114, 117, 120, 121, 125, 126, 127, 130, 131, 138, 141, 144, 150, 152, 153, 154, 159, 163, 166, 174, 175, 176, 177, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 218, 219, 220, 221, 222, 223, 228, 229, 231, 234, 235, 236, 237, 238, 241, 243, 244, 245, 246, 247, 249, 251, 252, 253, 255, 258, 261, 262, 264, 267, 270, 272, 273, 275, 276, 281}

 

[1, 82] = P[248], {7, 10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 62, 63, 65, 67, 70, 71, 73, 76, 77, 79, 81, 83, 85, 88, 90, 92, 93, 96, 97, 99, 102, 103, 105, 107, 108, 109, 110, 112, 113, 117, 118, 121, 122, 126, 127, 132, 133, 136, 138, 144, 147, 150, 156, 158, 162, 168, 171, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 215, 216, 217, 218, 219, 221, 222, 226, 227, 229, 230, 232, 234, 235, 237, 238, 240, 241, 243, 244, 246, 249, 251, 252, 253, 255, 258, 261, 262, 264, 267, 270, 271, 273, 275, 276, 280, 281}

 

[1, 84] = P[254], {7, 10, 14, 16, 19, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 51, 53, 55, 58, 59, 61, 63, 65, 68, 69, 71, 72, 73, 74, 76, 79, 81, 82, 85, 86, 91, 93, 94, 96, 97, 99, 101, 103, 108, 109, 110, 111, 113, 115, 118, 120, 124, 127, 132, 133, 135, 141, 144, 147, 148, 150, 151, 153, 154, 156, 159, 162, 165, 168, 169, 171, 172, 174, 175, 176, 177, 178, 179, 180, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 219, 220, 221, 222, 223, 224, 226, 228, 230, 231, 232, 234, 235, 239, 240, 242, 246, 247, 249, 250, 252, 255, 258, 261, 264, 267, 270, 273, 275, 276, 279, 281}

 

[1, 86] = P[260], {10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 52, 53, 55, 58, 60, 63, 64, 67, 68, 70, 71, 74, 75, 77, 79, 80, 81, 82, 85, 87, 89, 93, 94, 96, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 126, 127, 132, 133, 135, 141, 142, 144, 150, 151, 153, 159, 162, 163, 165, 166, 172, 174, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 219, 220, 221, 222, 223, 224, 228, 229, 231, 232, 234, 235, 236, 239, 240, 241, 243, 246, 247, 248, 249, 251, 253, 254, 255, 258, 261, 263, 264, 267, 270, 272, 273, 275, 276, 281}

 

[1, 88] = P[266], {12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 36, 37, 38, 40, 43, 46, 47, 49, 52, 53, 55, 57, 59, 61, 63, 65, 67, 68, 71, 72, 74, 76, 79, 80, 82, 84, 87, 88, 91, 93, 94, 96, 100, 102, 103, 104, 108, 109, 111, 113, 117, 119, 121, 123, 127, 128, 135, 141, 145, 147, 151, 152, 153, 157, 161, 162, 163, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 203, 204, 205, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 220, 222, 223, 225, 228, 229, 231, 233, 237, 238, 239, 240, 244, 246, 247, 252, 253, 254, 255, 258, 261, 263, 264, 267, 268, 269, 270, 273, 275, 276, 281}

 

[1, 90] = P[272], {10, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 35, 39, 40, 41, 43, 46, 49, 50, 51, 52, 54, 55, 58, 59, 60, 62, 65, 66, 68, 71, 73, 76, 77, 79, 80, 84, 85, 87, 90, 91, 94, 97, 99, 100, 101, 103, 106, 109, 110, 111, 112, 113, 114, 115, 117, 120, 121, 123, 129, 130, 132, 138, 144, 147, 148, 150, 153, 155, 162, 163, 168, 171, 172, 173, 174, 175, 176, 177, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 214, 216, 217, 219, 220, 222, 223, 224, 225, 228, 229, 230, 231, 232, 233, 234, 235, 237, 239, 240, 243, 244, 246, 247, 250, 251, 252, 254, 259, 261, 263, 264, 267, 268, 269, 270, 273, 276}

 

[1, 92] = P[278], {4, 7, 10, 12, 14, 15, 16, 19, 21, 22, 25, 28, 29, 31, 34, 36, 37, 39, 41, 43, 46, 47, 49, 51, 55, 56, 58, 59, 60, 64, 65, 67, 69, 71, 73, 74, 79, 81, 82, 83, 85, 87, 89, 91, 94, 96, 97, 100, 101, 104, 105, 107, 111, 114, 115, 116, 118, 120, 122, 124, 126, 130, 135, 138, 141, 144, 154, 157, 159, 168, 169, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 214, 216, 217, 219, 221, 223, 224, 225, 226, 228, 230, 231, 233, 237, 239, 240, 241, 243, 244, 246, 247, 249, 252, 253, 254, 255, 259, 261, 263, 264, 267, 270, 273, 275, 276, 279, 281, 282}

 

[1, 94] = P[284], {3, 4, 7, 10, 12, 14, 15, 16, 19, 22, 24, 25, 28, 30, 31, 34, 35, 36, 39, 41, 43, 46, 47, 49, 52, 53, 55, 58, 59, 61, 63, 66, 67, 70, 71, 73, 76, 78, 79, 82, 84, 85, 88, 90, 93, 95, 97, 99, 101, 103, 106, 108, 110, 112, 115, 118, 120, 121, 123, 127, 129, 135, 138, 139, 144, 145, 150, 156, 157, 159, 166, 168, 171, 174, 175, 176, 177, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 216, 217, 219, 220, 222, 223, 225, 226, 227, 228, 229, 231, 232, 234, 235, 237, 238, 239, 240, 241, 242, 243, 246, 247, 251, 252, 254, 255, 257, 258, 260, 261, 264, 267, 269, 270, 273, 274, 275, 279}

 

[1, 96] = P[290], {7, 10, 12, 13, 16, 17, 20, 21, 23, 25, 27, 29, 31, 32, 35, 37, 39, 43, 44, 46, 48, 49, 52, 53, 55, 58, 60, 61, 64, 65, 66, 67, 69, 70, 73, 75, 77, 80, 81, 84, 85, 88, 89, 91, 94, 96, 98, 101, 102, 104, 106, 109, 111, 112, 114, 116, 118, 121, 124, 126, 127, 129, 133, 135, 141, 142, 144, 150, 156, 161, 163, 166, 168, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 219, 220, 222, 224, 226, 229, 230, 231, 232, 234, 236, 237, 238, 239, 243, 244, 245, 246, 247, 249, 250, 251, 252, 255, 256, 258, 261, 264, 266, 267, 270, 273, 276, 279, 285}

 

[1, 98] = P[296], {8, 10, 13, 17, 19, 20, 22, 25, 27, 29, 31, 32, 34, 37, 38, 39, 41, 42, 43, 45, 46, 49, 51, 52, 54, 57, 58, 61, 63, 64, 67, 68, 73, 74, 76, 77, 78, 82, 83, 85, 88, 89, 92, 94, 95, 97, 101, 102, 106, 108, 109, 110, 112, 115, 118, 119, 120, 121, 124, 126, 127, 132, 135, 137, 138, 144, 145, 150, 156, 162, 165, 166, 169, 171, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 213, 214, 215, 216, 217, 218, 219, 220, 222, 223, 225, 226, 227, 228, 230, 231, 232, 233, 234, 236, 237, 240, 241, 245, 246, 247, 248, 249, 252, 253, 254, 258, 260, 264, 267, 269, 270, 273, 276, 279}

 

[1, 100] = P[302], {6, 8, 10, 13, 15, 17, 19, 20, 22, 25, 27, 29, 31, 32, 34, 37, 38, 39, 41, 42, 43, 45, 46, 49, 51, 52, 54, 57, 58, 61, 63, 64, 67, 68, 71, 72, 74, 76, 78, 80, 82, 84, 87, 88, 91, 92, 94, 96, 98, 100, 102, 104, 106, 109, 112, 113, 115, 117, 119, 121, 126, 128, 132, 136, 144, 150, 157, 165, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 216, 217, 219, 220, 222, 225, 226, 227, 228, 229, 230, 231, 233, 234, 235, 237, 239, 241, 243, 244, 246, 248, 252, 253, 254, 258, 259, 260, 264, 265, 267, 268, 270, 273, 276, 278, 279, 282}

 

numsols = 51

(7)

 


Download putnam2018.mw

 

Edit.
Maple can be also very useful in solving the difficult problem B6; it can be reduced to compute the (huge) coefficient of x^1842 in the polynomial

g := (1 + x + x^2 + x^3 + x^4 + x^5 + x^9)^2018;

The computation is very fast:

coeff(g, x, 1842):   evalf(%);
        
0.8048091229e1147

 

The Maple help contains a nice example of a puzzle solver named alphametic,  see  ?Iterator,Permute.
The goal is to determine the distinct digits represented by letters which satisfy a given  equation. The provided solved example is:

        "16540*781 = 12904836 + 12904"
i.e.  m=1, a=6, p=5 etc.

It's worth studying the program (written as a module) because it includes a few useful techniques.
I would suggest the following exercise for a Maple (young) programmer.

1.  When solving the "classical" puzzle  "FORTY+TEN+TEN=SIXTY", instead of the correct answer "29786+850+850=31486",   you will see
"2978Y+850+850=3148Y".
Try to find what's going on and correct the code.

2. The solutions given by alphametic include numbers beginning with a zero.
Execute e.g. .
Modify the code to produce only standard numbers.

 

I have recently visited the Queen's House at Greenwich  (see wiki),  an  important building in British architectural history (17th century).
I was impressed by the Great Hall Floor, whose central geometric decoration seems to be generated by a Maple program :-)

Here is my code for this. I hope you will like it.

restart;
with(plots): with(plottools):
n:=32: m:=3:#    n:=64: m:=7:

a[0], b[0] := exp(-Pi*I/n), exp(Pi*I/n):
c[0]:=b[0]+(a[0]-b[0])/sqrt(2)*exp(I*Pi/4):  
for k to m+1 do  
  c[k]:=a[k-1]+b[k-1]-c[k-1];
  b[k]:=c[k]*(1+exp(2*Pi*I/n))-b[k-1];
  a[k]:=conjugate(b[k])  od:
b[-1]:=c[0]*(1+exp(2*Pi*I/n))-b[0]:
a[-1]:=conjugate(b[-1]):
c[-1]:=a[-1]+b[-1]-c[0]:
seq( map[inplace](evalf@[Re,Im], w), w=[a,b,c] ):
Q:=polygonplot([seq([a[k],c[k],b[k],c[k+1]],k=0..m-1), [c[m],a[m],b[m]], [a[-1],b[-1],c[0]]]):
display(seq(rotate(Q, 2*k*Pi/n, [0,0]),k=0..n-1), disk([0,0],c[m][1]/3), axes=none, size=[800,800], color=black);

 

At a recent undegraduate competition the students had to compute the following limit

 

Limit( n * Diff( (exp(x)-1)/x, x$n), n=infinity ) assuming x<>0;

Limit(n*(Diff((exp(x)-1)/x, `$`(x, n))), n = infinity)

(1)

 

Maple is able to compute the symbolic n-fold derivative and I hoped that the limit will be computed at once.

Unfortunately it is not so easy.
Maybe someone finds a more more straightforward way.

 

restart;

f := n * diff( (exp(x)-1)/x, x$n );

n*(-1/x)^n*(-GAMMA(n+1)+GAMMA(n+1, -x))/x

(2)

limit(%, n=infinity);

limit(n*(-1/x)^n*(-GAMMA(n+1)+GAMMA(n+1, -x))/x, n = infinity)

(3)

simplify(%) assuming x>0;

limit(-(-1)^n*x^(-n-1)*n*(GAMMA(n+1)-GAMMA(n+1, -x)), n = infinity)

(4)

 

So, Maple cannot compute directly the limit.

 

convert(f, Int) assuming n::posint;

-n*(-1/x)^n*(-x)^(n+1)*GAMMA(2+n)*(Int(exp(_t1*x)*_t1^n, _t1 = 0 .. 1))/((n+1)*(Int(_k1^n*exp(-_k1), _k1 = 0 .. infinity))*x)

(5)

J:=simplify(%)  assuming n::posint;

n*(Int(exp(x*_k1)*_k1^n, _k1 = 0 .. 1))*GAMMA(n+1)/(Int(_k1^n*exp(-_k1), _k1 = 0 .. infinity))

(6)

L:=convert(J, Int) assuming n::posint;

n*(Int(exp(x*_k1)*_k1^n, _k1 = 0 .. 1))

(7)

L:=subs(_k1=u, L);

n*(Int(exp(x*u)*u^n, u = 0 .. 1))

(8)

 

Now it should be easy, but Maple needs help.

 

with(IntegrationTools):

L1:=Change(L, u^n = t, t) assuming n::posint;

Int(exp((x*t^(1/n)*n+ln(t))/n), t = 0 .. 1)

(9)

limit(L1, n=infinity);  # OK

exp(x)

(10)

####################################################################

Note that the limit can also be computed using an integration by parts, but Maple refuses to finalize:

Parts(L, exp(u*x)) assuming n::posint;

n*(exp(x)/(n+1)-(Int(u^(n+1)*x*exp(x*u)/(n+1), u = 0 .. 1)))

(11)

simplify(%);

n*(-x*(Int(u^(n+1)*exp(x*u), u = 0 .. 1))+exp(x))/(n+1)

(12)

limit(%, n=infinity);

limit(n*(-x*(Int(u^(n+1)*exp(x*u), u = 0 .. 1))+exp(x))/(n+1), n = infinity)

(13)

value(%);  # we are almost back!

limit(n*((-x)^(-n)*(-(n+1)*n*GAMMA(n)/x-(-x)^n*(x-n-1)*exp(x)/x+(n+1)*n*GAMMA(n, -x)/x)+exp(x))/(n+1), n = infinity)

(14)

 

add, floats, and Kahan sum

 

I found an intresting fact about the Maple command add for floating point values.
It seems that add in this case uses a summation algorithm in order to reduce the numerical error.
It is probably the Kahan summation algorithm (see wiki), but I wonder why this fact is not documented.

Here is a simple Maple procedure describing and implementing the algorithm.

 

 

restart;

Digits:=15;

15

(1)

KahanSum := proc(f::procedure, ab::range)  
local S,c,y,t, i;      # https://en.wikipedia.org/wiki/Kahan_summation_algorithm
S := 0.0;              # S = result (final sum: add(f(n), n=a..b))
c := 0.0;              # c = compensation for lost low-order bits.
for i from lhs(ab) to rhs(ab) do
    y := f(i) - c;     
    t := S + y;              
    c := (t - S) - y;        
    S := t;                  
od;                         
return S
end proc:

 

Now, a numerical example.

 

 

f:= n ->  evalf(1/(n+1/n^3+1) - 1/(n+1+1/(n+1)^3+1));

proc (n) options operator, arrow; evalf(1/(n+1/n^3+1)-1/(n+2+1/(n+1)^3)) end proc

(2)

n := 50000;
K := KahanSum(f, 1..n);

50000

 

.333313334133301

(3)

A := add(f(k),k=1..n);

.333313334133302

(4)

s:=0.0:  for i to n do s:=s+f(i) od:
's' = s;

s = .333313334133413

(5)

exact:=( 1/3 - 1/(n+1+1/(n+1)^3+1) );

6250249999999900000/18751875067501050009

(6)

evalf( [errK = K-exact, errA = A-exact, err_for=s-exact] );

[errK = 0., errA = 0.1e-14, err_for = 0.112e-12]

(7)

evalf[20]( [errK = K-exact, errA = A-exact, err_for=s-exact] );

[errK = -0.33461e-15, errA = 0.66539e-15, err_for = 0.11166539e-12]

(8)

 


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