cp's OEIS Frontend

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

%I A385234 #14 Jul 11 2025 15:35:17
%S A385234 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,
%T A385234 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,
%U A385234 9,8,8,11,8,11,10,10,13,9,14,12,13,15,12,17
%N A385234 a(n) is the number of partitions of n into primes of the form 4*k + 1.
%C A385234 a(0) = 1 corresponds to the empty partition {}.
%H A385234 Felix Huber, <a href="/A385234/b385234.txt">Table of n, a(n) for n = 0..10000</a>
%F A385234 G.f.: 1 / Product_{k>=1} (1-x^A002144(k)).
%F A385234 a(n) +  A385235(n) <= A000607(n) for n >= 1.
%F A385234 a(n) >= A024941(n).
%e A385234 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].
%p A385234 with(gfun):
%p A385234 A385234:=proc(N) # To get the first N terms.
%p A385234     local f,i,g,h,n;
%p A385234     f:=select(x->x mod 4=1,[seq(ithprime(i),i=1..NumberTheory:-pi(N))]);
%p A385234     g:=mul(1/(1-q^f[n]),n=1..nops(f)):
%p A385234     h:=series(g,q,N):
%p A385234     return op(seriestolist(h));
%p A385234 end proc;
%p A385234 A385234(84);
%t A385234 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]
%t A385234 (* _James C. McMahon_, Jul 11 2025 *)
%Y A385234 Cf. A000607, A002144, A024941, A024942, A319734, A385235.
%K A385234 nonn
%O A385234 0,31
%A A385234 _Felix Huber_, Jul 06 2025