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## Distribution of Integers Having a Divisor with an Odd Semi-prime

Maple 2021

Let N=pq be an odd semi-prime; What is the distribution of  integers that has a common divisor with N. We have shown that the distribution in [1,N-1] is a symmetric one, and there exsits a multiple of p lying to a multiple of q. We post the Maple source here.

gap := proc(a, b) return abs(a - b) - 1; end proc

HostsNdivisors := proc(N)

local i, j, g, d, L, s, t, m, p, q, P, Q, np, nq;

m := floor(1/2*N - 1/2);

L := evalf(sqrt(N));

P := Array();

Q := Array();

s := 1; t := 1;

for i from 3 to m do

d := gcd(i, N);

if 1 < d and d < L then P(s) := i; s++;

elif L < d then Q(t) := i; t++; end if;

end do;

np := s - 1;

nq := t - 1;

for i to np do printf("%3d,", P(i)); end do;

printf("\n");

for i to nq do printf("%3d,", Q(i)); end do;

printf("\n gaps: \n");

for i to np do

for j to nq do

p := P(i); q := Q(j);

g := gap(p, q);

printf("%4d,", g);

end do;

printf("\n");

end do;

end proc

HostOfpq := proc(p, q)

local alpha, s, t, g, r, S, T, i, j;

S := 1/2*q - 1/2;

T := 1/2*p - 1/2;

alpha := floor(q/p);

r := q - alpha*p;

for s to S do

for t to T do

g := abs((t*alpha - s)*p + t*r) - 1;

printf("%4d,", g);

end do;

printf("\n");

end do;

end proc

MapleSource.pdf

MapleSource.mw

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