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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).

%I A248956 #28 Jun 25 2022 21:43:58
%S A248956 1,3,5,9,13,19,27,37,49,65,85,109,139,175,219,273,337,413,505,613,741,
%T A248956 893,1071,1279,1523,1807,2137,2521,2965,3477,4069,4749,5529,6425,7449,
%U A248956 8619,9955,11475,13203,15167,17393,19913,22765,25985,29617,33713,38321,43501
%N 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).
%C A248956 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.
%H A248956 Hiroaki Yamanouchi, <a href="/A248956/b248956.txt">Table of n, a(n) for n = 0..1000</a>
%F A248956 a(n) = -1 + 2*Sum_{k=0..n} a*(k) where a*(n) = A000009(n).
%F A248956 a(n) = A248955(p^n), where p is any prime. - _Michel Marcus_, Nov 07 2014
%F A248956 a(n) = 2*A036469(n) - 1. - _Hiroaki Yamanouchi_, Nov 21 2014
%e A248956 a(1) = 3: -p*x+p; -x+p; x^2 - (p+1)*x + p.
%Y A248956 Cf. A248955, A248348.
%Y A248956 Partial sums of A087135.
%K A248956 nonn
%O A248956 0,2
%A A248956 _Reiner Moewald_, Oct 17 2014
%E A248956 a(20)-a(22) from _Michel Marcus_, Nov 07 2014
%E A248956 a(23)-a(47) from _Hiroaki Yamanouchi_, Nov 21 2014