A024942 -id:A024942 - cp's OEIS frontend

A024941 Number of partitions of n into distinct primes of the form 4k + 1.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 2, 3, 1, 1, 1, 2, 1, 0, 3, 2, 0, 0, 1, 5, 1, 0, 1, 2, 3, 1, 3, 3, 1, 0, 2, 5, 3, 1, 1, 2, 3, 2, 4, 4, 1, 1, 2, 7, 4, 1, 2, 3

Offset: 0

Clark Kimberling

nonn

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

			a(41) = 1 since it can be expressed as a sum of primes of the form 4k + 1 in only one way: a trivial partition containing just itself.
a(42) = 2 since 42 = 5 + 37 = 13 + 29.
Although 43 = 2 * 13 + 17 = 6 * 5 + 13, none of those consist of distinct primes only. Hence a(43) = 0.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A024942 (4k - 1).

searchMax = 120; primes4kp1 = Select[4Range[Floor[searchMax/4]] + 1, PrimeQ]; Table[Length[Select[IntegerPartitions[n, All, primes4kp1], DuplicateFreeQ]], {n, 0, searchMax}] (* Alonso del Arte, Apr 17 2019 *)

{ my(V=select(x->x%4==1,primes(40))); my(x='x+O('x^V[#V])); Vec(prod(k=1,#V,1+x^V[k])) } \\ Joerg Arndt, Apr 19 2019

Definition clarified by Felix Fröhlich, Apr 17 2019

a(0) = 1 prepended by Joerg Arndt, Apr 19 2019

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

Original entry on oeis.org

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

Felix Huber, Jul 06 2025

nonn

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

			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].

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

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

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);

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 *)

G.f.: 1 / Product_{k>=1} (1-x^A002144(k)).
a(n) +  A385235(n) <= A000607(n) for n >= 1.
a(n) >= A024941(n).

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

Original entry on oeis.org

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

Felix Huber, Jul 06 2025

nonn

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

			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].

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

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

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);

G.f.: 1 / Product_{k>=1} (1-x^A002145(k)).
a(n) + A385234(n) <= A000607(n) for n >= 1.
a(n) >= A024942(n).

cp's OEIS Frontend

A024941 Number of partitions of n into distinct primes of the form 4k + 1.

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

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

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Maple

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