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

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, 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, 34, 35, 37, 39, 41, 44, 46, 49, 51, 55, 57, 59, 64, 66, 70, 73, 77

Offset

0,15

Comments

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

Links

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

Formula

G.f.: 1 / Product_{k>=1} (1-x^A002145(k)).

a(n) + A385234(n) <= A000607(n) for n >= 1.

a(n) >= A024942(n).

Example

The a(14) = 2 partitions of 14 into primes of the form 4*k + 3 are [3, 11] and [7, 7].
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].

Maple

with(gfun):
A385235:=proc(N) # To get the first N terms.
    local f, i, g, h, n;
    f:=select(x->x mod 4=3, [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;
A385235(76);

Crossrefs

Cf. A000607, A002145, A024941, A024942, A385234.

Sequence in context: A298604 A190353 A331904 * A025829 A029285 A134337

Adjacent sequences: A385232 A385233 A385234 * A385236 A385237 A385238

Keyword

nonn

Author

Felix Huber, Jul 06 2025

Status

approved