A341949 - OEIS

%I #11 Feb 15 2022 08:15:52

%S 1,1,2,2,4,4,7,6,9,8,12,10,16,12,19,15,24,18,29,21,35,25,41,29,49,33,

%T 56,37,63,41,72,46,82,51,91,58,105,63,115,68,128,77,143,83,158,90,174,

%U 101,193,107,211,116,231,128,250,134,273,142,294,157,321,165,347,176,374

%N Number of partitions of n into 6 primes (counting 1 as a prime).

%p b:= proc(n, i) option remember; series(`if`(n=0, 1,

%p      `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(

%p      `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 7)

%p     end:

%p a:= n-> coeff(b(n, numtheory[pi](n)), x, 6):

%p seq(a(n), n=6..70);  # _Alois P. Heinz_, Feb 24 2021

%t b[n_, i_] := b[n, i] = Series[If[n == 0, 1,

%t      If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][

%t      If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 7}];

%t a[n_] := Coefficient[b[n, PrimePi[n]], x, 6];

%t Table[a[n], {n, 6, 70}] (* _Jean-François Alcover_, Feb 15 2022, after _Alois P. Heinz_ *)

%Y Cf. A008578, A034891, A259196, A341945, A341946, A341947, A341948, A341950, A341951.

%K nonn

%O 6,3

%A _Ilya Gutkovskiy_, Feb 24 2021