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%I #14 Jan 11 2021 10:08:37
%S 1,0,0,0,1,1,1,1,2,2,3,2,4,4,6,5,8,7,11,10,15,13,20,17,26,23,34,29,43,
%T 38,55,49,69,62,88,78,109,97,135,122,167,150,205,186,251,227,306,277,
%U 371,337,448,407,539,492,647,591,773,707,922,845,1096,1005,1298,1193,1535,1412,1809,1667,2127
%N Number of partitions of n into an even number of primes.
%H Vaclav Kotesovec, <a href="/A184198/b184198.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = (A000607(n)+A048165(n))/2.
%e n=18 can be partitioned in A000607(18)=19 ways into primes, of which a(18)=11 are even, namely 11+7, 13+5, 5+5+5+3, 7+5+3+3, 3+3+3+3+3+3, 7+7+2+2, 11+3+2+2, 5+3+3+3+2+2, 5+5+2+2+2+2, 7+3+2+2+2+2, 3+3+2+2+2+2+2+2.
%e The remaining A184199(18)=8 are odd.
%t Table[Count[IntegerPartitions[n],_?(AllTrue[#,PrimeQ]&&EvenQ[Length[ #]]&)], {n,0,70}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Feb 16 2018 *)
%o (PARI)
%o parts(n, pred, y)={prod(k=1, n, if(pred(k), 1/(1-y*x^k) + O(x*x^n), 1))}
%o {my(n=80); Vec(parts(n, isprime, 1) + parts(n, isprime, -1))/2} \\ _Andrew Howroyd_, Dec 28 2017
%Y Cf. A000607, A048165, A184199, A184171.
%K nonn,easy
%O 0,9
%A _R. J. Mathar_, Jan 10 2011
%E a(31)-a(69) corrected by _Andrew Howroyd_, Dec 28 2017
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