cp's OEIS Frontend

A307726 Number of partitions of n into 2 prime powers (not including 1).

%I A307726 #12 Apr 29 2019 21:03:01
%S A307726 0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,2,4,3,4,4,4,2,4,3,4,4,4,3,5,3,6,4,
%T A307726 7,4,7,2,5,4,6,3,5,3,5,5,6,2,7,3,7,4,6,2,8,3,7,4,6,2,7,3,6,4,7,2,9,2,
%U A307726 7,5,7,2,9,3,7,6,7,3,9,2,8,4,6,4,10,3,9,4,7,3,11,4,8,3,7,2,10,2,8,3,8
%N A307726 Number of partitions of n into 2 prime powers (not including 1).
%H A307726 Robert Israel, <a href="/A307726/b307726.txt">Table of n, a(n) for n = 0..10000</a>
%H A307726 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A307726 a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^A246655(k)).
%e A307726 a(10) = 3 because we have [8, 2], [7, 3] and [5, 5].
%p A307726 # note that this requires A246655 to be pre-computed
%p A307726 f:= proc(n, k, pmax) option remember;
%p A307726   local t, p, j;
%p A307726   if n = 0 then return `if`(k=0, 1, 0) fi;
%p A307726   if k = 0 then return 0 fi;
%p A307726   if n > k*pmax then return 0 fi;
%p A307726   t:= 0:
%p A307726   for p in A246655 do
%p A307726     if p > pmax then return t fi;
%p A307726     t:= t + add(procname(n-j*p, k-j, min(p-1, n-j*p)), j=1..min(k, floor(n/p)))
%p A307726   od;
%p A307726   t
%p A307726 end proc:
%p A307726 map(f, [$0..100]); # _Robert Israel_, Apr 29 2019
%t A307726 Array[Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimePowerQ] &)] &, 101, 0]
%Y A307726 Cf. A000961, A023894, A061358, A071068, A071330, A071331, A246655, A280242, A307727.
%K A307726 nonn,look
%O A307726 0,7
%A A307726 _Ilya Gutkovskiy_, Apr 24 2019