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A341946 Number of partitions of n into 3 primes (counting 1 as a prime).

%I A341946 #15 Feb 14 2022 10:36:49
%S A341946 1,1,2,2,3,2,4,2,4,2,4,2,6,3,6,2,6,3,8,3,8,3,9,4,10,3,9,2,10,4,12,3,
%T A341946 12,4,13,4,13,3,14,3,15,5,16,4,17,4,18,6,19,4,19,3,20,6,20,3,20,4,23,
%U A341946 7,23,4,26,5,26,6,23,3,27,5,28,7,28,6,33,5,31,7,30,5,34
%N A341946 Number of partitions of n into 3 primes (counting 1 as a prime).
%H A341946 Alois P. Heinz, <a href="/A341946/b341946.txt">Table of n, a(n) for n = 3..10000</a>
%p A341946 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p A341946      `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
%p A341946      `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 4)
%p A341946     end:
%p A341946 a:= n-> coeff(b(n, numtheory[pi](n)), x, 3):
%p A341946 seq(a(n), n=3..83);  # _Alois P. Heinz_, Feb 24 2021
%t A341946 b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
%t A341946      If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
%t A341946      If[i == 0, 1, Prime[i]]] + b[n, i-1]]], {x, 0, 4}];
%t A341946 a[n_] := Coefficient[b[n, PrimePi[n]], x, 3];
%t A341946 Table[a[n], {n, 3, 83}] (* _Jean-François Alcover_, Feb 14 2022, after _Alois P. Heinz_ *)
%Y A341946 Cf. A008578, A034891, A068307, A341945, A341947, A341948, A341949, A341950, A341951.
%K A341946 nonn,look
%O A341946 3,3
%A A341946 _Ilya Gutkovskiy_, Feb 24 2021