A385234 a(n) is the number of partitions of n into primes of the form 4*k + 1.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 3, 2, 2, 3, 2, 3, 2, 3, 4, 2, 3, 3, 4, 5, 3, 5, 4, 5, 5, 5, 6, 5, 6, 5, 7, 7, 6, 8, 7, 9, 8, 8, 11, 8, 11, 10, 10, 13, 9, 14, 12, 13, 15, 12, 17

Offset: 0

Author

Felix Huber, Jul 06 2025

Keywords

• nonn

Comments

a(0) = 1 corresponds to the empty partition {}.

Examples

			The a(53) = 3 partitions of 53 into primes of the form 4*k + 1 are [53], [5, 5, 13, 13, 17] and [5, 5, 5, 5, 5, 5, 5, 5, 13].
		

Links

• Felix Huber, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000607, A002144, A024941, A024942, A319734, A385235.

Programs

Maple

with(gfun):
A385234:=proc(N) # To get the first N terms.
    local f,i,g,h,n;
    f:=select(x->x mod 4=1,[seq(ithprime(i),i=1..NumberTheory:-pi(N))]);
    g:=mul(1/(1-q^f[n]),n=1..nops(f)):
    h:=series(g,q,N):
    return op(seriestolist(h));
end proc;
A385234(84);

Mathematica

A385234[N_]:=Module[{f, g},f = Select[Prime[Range[PrimePi[N]]], Mod[#, 4] == 1 &]; g = Product[1/(1 - q^f[[n]]),{n, 1, Length[f]}];CoefficientList[Series[g, {q, 0, N}], q]];A385234[83]
(* James C. McMahon, Jul 11 2025 *)

Formula

G.f.: 1 / Product_{k>=1} (1-x^A002144(k)).

a(n) + A385235(n) <= A000607(n) for n >= 1.

a(n) >= A024941(n).