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A348588 Number of partitions of n into 3 parts with at least 1 prime part.

%I A348588 #11 Nov 10 2021 16:59:25
%S A348588 0,0,0,0,1,2,2,4,4,6,7,8,10,12,13,17,17,19,22,25,26,31,31,36,37,42,43,
%T A348588 49,49,56,56,64,63,72,70,80,79,87,87,99,94,107,105,116,114,126,123,
%U A348588 139,134,148,145,158,155,173,166,184,178,195,189,211,202,222,215,236,226,250,242
%N A348588 Number of partitions of n into 3 parts with at least 1 prime part.
%H A348588 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A348588 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} sign(c(i) + c(j) + c(n-i-j)), where c is the prime characteristic (A010051).
%p A348588 b:= proc(n, i, t) option remember; series(
%p A348588      `if`(n=0, t, `if`(i<1, 0, expand(x*b(n-i, min(n-i, i),
%p A348588      `if`(isprime(i), 1, t)))+b(n, i-1, t))), x, 4)
%p A348588     end:
%p A348588 a:= n-> coeff(b(n$2, 0), x, 3):
%p A348588 seq(a(n), n=0..66);  # _Alois P. Heinz_, Oct 24 2021
%t A348588 Table[Length@Select[IntegerPartitions[k,{3}],Or@@PrimeQ@#&],{k,0,66}] (* _Giorgos Kalogeropoulos_, Oct 24 2021 *)
%Y A348588 Cf. A010051, A068307, A235945, A340572.
%K A348588 nonn
%O A348588 0,6
%A A348588 _Wesley Ivan Hurt_, Oct 24 2021