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A341974 Number of partitions of n into 3 distinct primes (counting 1 as a prime).

%I A341974 #13 Jul 13 2021 02:51:37
%S A341974 1,0,1,1,2,1,1,1,2,2,2,2,2,3,3,4,3,4,2,5,4,5,2,5,2,7,4,6,3,8,3,9,4,7,
%T A341974 2,9,3,10,5,9,4,12,3,13,6,12,4,14,3,16,6,13,3,16,3,19,7,14,3,19,5,21,
%U A341974 6,15,3,23,5,23,7,18,5,26,5,26,7,21,5,29,4,28,9,25,4,30,4
%N A341974 Number of partitions of n into 3 distinct primes (counting 1 as a prime).
%H A341974 Alois P. Heinz, <a href="/A341974/b341974.txt">Table of n, a(n) for n = 6..10000</a>
%p A341974 b:= proc(n, i) option remember; series(`if`(n=0, 1,
%p A341974      `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i-1)))(
%p A341974      `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 4)
%p A341974     end:
%p A341974 a:= n-> coeff(b(n, numtheory[pi](n)), x, 3):
%p A341974 seq(a(n), n=6..90);  # _Alois P. Heinz_, Feb 24 2021
%t A341974 b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
%t A341974      If[i<0, 0, Function[p, If[p>n, 0, x*b[n-p, i-1]]][
%t A341974      If[i == 0, 1, Prime[i]]] + b[n, i-1]]], {x, 0, 4}];
%t A341974 a[n_] := Coefficient[b[n, PrimePi[n]], x, 3];
%t A341974 Table[a[n], {n, 6, 1000}] (* _Jean-François Alcover_, Jul 13 2021, after _Alois P. Heinz_ *)
%Y A341974 Cf. A008578, A036497, A125688, A341946, A341973, A341975, A341976, A341977.
%K A341974 nonn
%O A341974 6,5
%A A341974 _Ilya Gutkovskiy_, Feb 24 2021