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A385235 a(n) is the number of partitions of n into primes of the form 4*k + 3.

%I A385235 #6 Jul 10 2025 23:51:11
%S A385235 1,0,0,1,0,0,1,1,0,1,1,1,1,1,2,1,1,2,2,2,2,3,3,3,3,4,4,3,5,5,5,6,6,7,
%T A385235 7,7,8,9,9,9,11,11,12,13,14,15,15,17,17,19,20,20,23,24,25,26,29,30,30,
%U A385235 34,35,37,39,41,44,46,49,51,55,57,59,64,66,70,73,77
%N A385235 a(n) is the number of partitions of n into primes of the form 4*k + 3.
%C A385235 a(0) = 1 corresponds to the empty partition {}.
%H A385235 Felix Huber, <a href="/A385235/b385235.txt">Table of n, a(n) for n = 0..10000</a>
%F A385235 G.f.: 1 / Product_{k>=1} (1-x^A002145(k)).
%F A385235 a(n) + A385234(n) <= A000607(n) for n >= 1.
%F A385235 a(n) >= A024942(n).
%e A385235 The a(14) = 2 partitions of 14 into primes of the form 4*k + 3 are [3, 11] and [7, 7].
%e A385235 The a(23) = 3 partitions of 23 into primes of the form 4*k + 3 are [23], [3, 3, 3, 3, 11] and [3, 3, 3, 7, 7].
%p A385235 with(gfun):
%p A385235 A385235:=proc(N) # To get the first N terms.
%p A385235     local f,i,g,h,n;
%p A385235     f:=select(x->x mod 4=3,[seq(ithprime(i),i=1..NumberTheory:-pi(N))]);
%p A385235     g:=mul(1/(1-q^f[n]),n=1..nops(f)):
%p A385235     h:=series(g,q,N):
%p A385235     return op(seriestolist(h));
%p A385235 end proc;
%p A385235 A385235(76);
%Y A385235 Cf. A000607, A002145, A024941, A024942, A385234.
%K A385235 nonn
%O A385235 0,15
%A A385235 _Felix Huber_, Jul 06 2025