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Number of partitions of n into 2 prime powers (not including 1).
4

%I #12 Apr 29 2019 21:03:01

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

%H Robert Israel, <a href="/A307726/b307726.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^A246655(k)).

%e a(10) = 3 because we have [8, 2], [7, 3] and [5, 5].

%p # note that this requires A246655 to be pre-computed

%p f:= proc(n, k, pmax) option remember;

%p local t, p, j;

%p if n = 0 then return `if`(k=0, 1, 0) fi;

%p if k = 0 then return 0 fi;

%p if n > k*pmax then return 0 fi;

%p t:= 0:

%p for p in A246655 do

%p if p > pmax then return t fi;

%p t:= t + add(procname(n-j*p, k-j, min(p-1, n-j*p)), j=1..min(k, floor(n/p)))

%p od;

%p t

%p end proc:

%p map(f, [$0..100]); # _Robert Israel_, Apr 29 2019

%t Array[Count[IntegerPartitions[#, {2}], _?(AllTrue[#, PrimePowerQ] &)] &, 101, 0]

%Y Cf. A000961, A023894, A061358, A071068, A071330, A071331, A246655, A280242, A307727.

%K nonn,look

%O 0,7

%A _Ilya Gutkovskiy_, Apr 24 2019