A248955 Number of polynomials a_k*x^k + ... + a_1*x + n with k > 0, integer coefficients and distinct positive integer roots and positive integers n.

1, 3, 3, 5, 3, 9, 3, 9, 5, 9, 3, 17, 3, 9, 9, 13, 3, 17, 3, 17, 9, 9, 3, 31, 5, 9, 9, 17, 3, 29, 3, 19, 9, 9, 9, 35, 3, 9, 9, 31, 3, 29, 3, 17, 17, 9, 3, 49, 5, 17, 9, 17, 3, 31, 9, 31, 9, 9, 3, 61, 3, 9, 17, 27, 9, 29, 3, 17, 9, 29, 3, 67, 3, 9, 17, 17, 9

Offset: 1

Author

Reiner Moewald, Oct 17 2014

Keywords

• nonn

Comments

If D_n is the set of all positive divisors of n, then a(n) gives the number of all subsets of D_n for which the product of all their elements is a divisor of n. a(n) depends only on the prime signature of n.

Examples

			a(2) = 3: -2x+2; -x+2; x^2 - 3x + 2.
		

Links

• Reiner Moewald, Table of n, a(n) for n = 1..2159

Crossrefs

Cf. A248410, A248956.

Programs

PARI

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {d = divisors(n); nbd = #d; nbts = 2^nbd-1; nbs = 0; for (i=1, nbts, bin = padbin(i, nbd); prd = prod(j=1, nbd,  if (bin[j], d[j], 1)); if (n % prd == 0, nbs++);); nbs;} \\ Michel Marcus, Nov 07 2014

Formula

a(p) = 3, for p prime. - Michel Marcus, Nov 07 2014