A348588 - OEIS

%I #11 Nov 10 2021 16:59:25

%S 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 49,49,56,56,64,63,72,70,80,79,87,87,99,94,107,105,116,114,126,123,

%U 139,134,148,145,158,155,173,166,184,178,195,189,211,202,222,215,236,226,250,242

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

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

%F 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 b:= proc(n, i, t) option remember; series(

%p      `if`(n=0, t, `if`(i<1, 0, expand(x*b(n-i, min(n-i, i),

%p      `if`(isprime(i), 1, t)))+b(n, i-1, t))), x, 4)

%p     end:

%p a:= n-> coeff(b(n$2, 0), x, 3):

%p seq(a(n), n=0..66);  # _Alois P. Heinz_, Oct 24 2021

%t Table[Length@Select[IntegerPartitions[k,{3}],Or@@PrimeQ@#&],{k,0,66}] (* _Giorgos Kalogeropoulos_, Oct 24 2021 *)

%Y Cf. A010051, A068307, A235945, A340572.

%K nonn

%O 0,6

%A _Wesley Ivan Hurt_, Oct 24 2021