A248956 - OEIS

%I #28 Jun 25 2022 21:43:58

%S 1,3,5,9,13,19,27,37,49,65,85,109,139,175,219,273,337,413,505,613,741,

%T 893,1071,1279,1523,1807,2137,2521,2965,3477,4069,4749,5529,6425,7449,

%U 8619,9955,11475,13203,15167,17393,19913,22765,25985,29617,33713,38321,43501

%N 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 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 Hiroaki Yamanouchi, <a href="/A248956/b248956.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = A248955(p^n), where p is any prime. - _Michel Marcus_, Nov 07 2014

%F a(n) = 2*A036469(n) - 1. - _Hiroaki Yamanouchi_, Nov 21 2014

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

%Y Cf. A248955, A248348.

%Y Partial sums of A087135.

%K nonn

%O 0,2

%A _Reiner Moewald_, Oct 17 2014

%E a(20)-a(22) from _Michel Marcus_, Nov 07 2014

%E a(23)-a(47) from _Hiroaki Yamanouchi_, Nov 21 2014