A340728 a(n) is the number of divisors d of n such that n/d - d is prime.

0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 1, 0, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 0, 0, 1, 0, 3, 1

Offset: 1

Author

J. M. Bergot and Robert Israel, Jan 17 2021

Keywords

• nonn

Comments

If n is odd, then a(n) = 0 unless n is in A000466, in which case a(n) = 1.

Examples

			a(8) = 2; among the divisors {1,2,4,8} of 8, there are two cases where 8/d-d is prime: 8/1-1 = 7 and 8/2-2 = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000466, A010051, A179993, A340729.

Programs

Maple

f:= proc(n) local D,i,m;
  D:= sort(convert(numtheory:-divisors(n),list));
  m:= nops(D);
  nops(select(i -> isprime(D[m+1-i]-D[i]), [$1..(m+1)/2]));
end proc:
map(f, [$1..100]);

PARI

a(n) = sumdiv(n, d, isprime(n/d-d)); \\ Michel Marcus, Jan 18 2021

Formula

a(n) = Sum_{d|n} c(n/d-d), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Jan 18 2021