A024941 -id:A024941 - cp's OEIS frontend

A024942 Number of partitions of n into distinct primes of the form 4k + 3.

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 2, 1, 0, 1, 2, 2, 1, 1, 2, 1, 1, 0, 2, 3, 0, 2, 3, 3, 0, 1, 4, 1, 1, 2, 4, 2, 1, 3, 3, 3, 1, 3, 4, 3, 2, 2, 6, 3, 1, 4, 6, 4, 1, 6, 6, 3, 3, 5, 7, 3, 3, 6, 7, 6, 2, 7, 9, 3, 2, 7, 10, 2, 4, 11, 10, 4, 5, 12, 7

Offset: 0

Clark Kimberling

nonn

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

			a(26) = 2 since 26 = 3 + 23 = 7 + 19.
Even though 27 = (3 * 3) + 7 + 11 = (2 * 3) + (3 * 7) = (9 * 3), there is no partition of 27 into primes of the form 4k - 1 with all parts distinct. Hence a(27) = 0.

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

Cf. A024941 (4k + 1).

searchMax = 108; primes4km1 = Select[4Range[Ceiling[searchMax/4]] - 1, PrimeQ]; Table[Length[Select[IntegerPartitions[n, All, primes4km1], DuplicateFreeQ]], {n, 0, searchMax}] (* Alonso del Arte, Apr 16 2019 *)

{ my(V=select(x->x%4==3,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

A282970 Number of partitions of n into primes of form x^2 + y^2 (A002313).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 12, 12, 13, 14, 14, 17, 16, 19, 19, 21, 22, 23, 25, 27, 27, 30, 30, 34, 35, 37, 40, 41, 45, 46, 50, 52, 55, 58, 60, 65, 67, 71, 75, 78, 84, 86, 92, 97, 100, 108, 110, 118, 123, 127, 137, 139, 150, 154, 162

Offset: 0

Ilya Gutkovskiy, Feb 25 2017

nonn

Number of partitions of n into primes congruent to 1 or 2 mod 4.

			a(10) = 2 because we have [5, 5] and [2, 2, 2, 2, 2].

Index entries for related partition-counting sequences

Cf. A000607, A002313, A024941, A281273.

nmax = 82; CoefficientList[Series[Product[1/(1 - Boole[SquaresR[2, k] != 0 && PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Vec(prod(k=1, 82, (1/(1 - (isprime(k) && k%4<3)*x^k))) + O(x^83)) \\ Indranil Ghosh, Mar 15 2017

G.f.: Product_{k>=1} 1/(1 - x^A002313(k)).

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).

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A024942 Number of partitions of n into distinct primes of the form 4k + 3.

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A282970 Number of partitions of n into primes of form x^2 + y^2 (A002313).

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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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