A248956 Number of polynomials a_k*x^k + ... + a_1*x + a_0 with k > 0, integer coefficients and only non-multiple positive integer roots and a_0 = p^n (p is a prime).

1, 3, 5, 9, 13, 19, 27, 37, 49, 65, 85, 109, 139, 175, 219, 273, 337, 413, 505, 613, 741, 893, 1071, 1279, 1523, 1807, 2137, 2521, 2965, 3477, 4069, 4749, 5529, 6425, 7449, 8619, 9955, 11475, 13203, 15167, 17393, 19913, 22765, 25985, 29617, 33713, 38321, 43501

Offset

0,2

Comments

If D_n = {p^0, ..., p^n} is the set of all positive divisors of p^n (p is a prime), then a(n) gives the number of all subsets of D_n for which the product of all their elements is a divisor of p^n. Furthermore, a(n) gives the number of all strict partitions of n including the integer 0.

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..1000

Formula

a(n) = -1 + 2*Sum_{k=0..n} a*(k) where a*(n) = A000009(n).

a(n) = A248955(p^n), where p is any prime. - Michel Marcus, Nov 07 2014

a(n) = 2*A036469(n) - 1. - Hiroaki Yamanouchi, Nov 21 2014

Example

a(1) = 3: -p*x+p; -x+p; x^2 - (p+1)*x + p.

Crossrefs

Cf. A248955, A248348.

Partial sums of A087135.

Sequence in context: A213207 A102378 A118026 * A108754 A033499 A267262

Adjacent sequences: A248953 A248954 A248955 * A248957 A248958 A248959

Keyword

nonn

Author

Reiner Moewald, Oct 17 2014

Extensions

a(20)-a(22) from Michel Marcus, Nov 07 2014

a(23)-a(47) from Hiroaki Yamanouchi, Nov 21 2014

Status

approved