A237281 -id:A237281 - cp's OEIS frontend

A237278 Numbers k such that A000041(k) == 0 (mod 4).

11, 15, 21, 26, 30, 55, 58, 59, 62, 66, 70, 74, 75, 78, 80, 84, 94, 96, 98, 100, 106, 108, 109, 112, 113, 116, 117, 120, 122, 124, 125, 126, 128, 130, 131, 133, 135, 136, 137, 141, 142, 149, 153, 154, 171, 179, 180, 187, 191, 200, 205, 206, 230, 231, 236

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(11) = 56 == 0 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237279, A237280, A237281, A243935 (see crossrefs).

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

is(n)=numbpart(n)%4==0 \\ Charles R Greathouse IV, Apr 08 2015

A237276 Numbers k such that A000041(k) == 1 (mod 3).

Original entry on oeis.org

0, 1, 5, 8, 18, 19, 23, 27, 28, 34, 36, 37, 44, 45, 50, 54, 55, 59, 62, 64, 72, 73, 77, 81, 82, 86, 89, 91, 95, 98, 99, 100, 110, 112, 113, 116, 117, 118, 119, 122, 128, 134, 137, 139, 140, 143, 146, 148, 149, 150, 152, 154, 155, 157, 158, 161, 166, 168, 170

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by A083214, A237276, and A237277.

			A000041(8) = 22 == 1 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Clark Kimberling)

Cf. A000041, A083214, A237276, A237277.

f[n_, k_] := Select[Range[0, 250], Mod[PartitionsP[#], n] == k &]
Table[f[3, k], {k, 0, 2}] (* A083214, A237276, A237277 *)
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

a(1)=0 inserted by Amiram Eldar, May 22 2025

A237280 Numbers k such that A000041(k) == 2 (mod 4).

Original entry on oeis.org

2, 8, 9, 10, 19, 22, 25, 27, 28, 31, 34, 40, 42, 45, 46, 47, 50, 57, 64, 65, 79, 86, 97, 101, 103, 110, 129, 147, 151, 158, 160, 163, 167, 170, 174, 175, 176, 184, 197, 198, 207, 213, 217, 224, 227, 228, 241, 245, 246, 247

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(8) = 22 == 2 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237278, A237279, A237281.

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

A121062 Partition numbers mod 4.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 3, 3, 2, 2, 2, 0, 1, 1, 3, 0, 3, 1, 1, 2, 3, 0, 2, 3, 3, 2, 0, 2, 2, 1, 0, 2, 1, 3, 2, 3, 1, 1, 3, 1, 2, 3, 2, 1, 3, 2, 2, 2, 1, 1, 2, 3, 1, 3, 3, 0, 3, 2, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 3, 1, 0, 1, 3, 1, 0, 0, 3, 3, 0, 2, 0, 3, 3, 1, 0, 1, 2, 1, 1, 1, 1, 3, 3, 1, 0, 3, 0, 2, 0, 3, 0, 2, 3, 2, 1

Offset: 0

Jonathan Vos Post, Aug 10 2006

P_n==0: 11, 15, 21, 26, 30, 55, 58, 59, 62, 66, 70, 74, 75, 78, 80, 84, 94, 96, 98, 100, ... A237278.
P_n==1: 0, 1, 4, 12, 13, 17, 18, 29, 32, 36, 37, 39, 43, 48, 49, 52, 61, 67, 69, 71, 73, ... A237279.
P_n==2: 2, 8, 9, 10, 19, 22, 25, 27, 28, 31, 34, 40, 42, 45, 46, 47, 50, 57, 64, 65, 79, ... A237280.
P_n==3: 3, 5, 6, 7, 14, 16, 20, 23, 24, 33, 35, 38, 41, 44, 51, 53, 54, 56, 60, 63, 68, ... A237281.

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

Cf. A000041, A010873, A049575.

Cf. A237278, A237279, A237280, A237281.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     (b(n, i-1)+b(n-i, min(n-i, i))) mod 4))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..104);  # Alois P. Heinz, Dec 20 2024

f[n_] := Mod[PartitionsP@n, 4]; Table[f@n, {n, 0, 104}] (* Robert G. Wilson v *)

a(n) = numbpart(n) % 4; \\ Michel Marcus, Jun 29 2016

a(n) = A000041(n) mod 4 = A010873(A000041(n)).

a(n) = A000025(n) mod 4. - John M. Campbell, Jun 29 2016

More terms from Robert G. Wilson v, Aug 17 2006

A237279 Numbers k such that A000041(k) == 1 (mod 4).

Original entry on oeis.org

1, 4, 12, 13, 17, 18, 29, 32, 36, 37, 39, 43, 48, 49, 52, 61, 67, 69, 71, 73, 83, 85, 87, 88, 89, 90, 93, 104, 105, 107, 114, 119, 123, 132, 143, 144, 145, 148, 150, 152, 155, 157, 162, 172, 181, 185, 186, 189, 190, 193, 194, 195, 199, 203, 208, 216, 219

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(4) = 5 == 1 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237278, A237280, A237281.

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

A237277 Numbers k such that A000041(k) == 2 (mod 3).

Original entry on oeis.org

2, 4, 6, 11, 12, 13, 15, 25, 29, 31, 38, 42, 47, 49, 56, 58, 60, 65, 66, 67, 69, 74, 76, 78, 79, 83, 84, 85, 87, 90, 92, 93, 94, 96, 101, 103, 105, 108, 109, 114, 120, 121, 123, 126, 127, 130, 131, 132, 135, 136, 141, 144, 145, 153, 156, 159, 165, 167, 171

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by A083214, A237276, and A237277.

			A000041(6) = 11 == 2 (mod 3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A083214, A237276.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[3, k], {k, 0, 2}] (* A083214, A237276, A237277 *)
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

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A237278 Numbers k such that A000041(k) == 0 (mod 4).

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A237276 Numbers k such that A000041(k) == 1 (mod 3).

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A237280 Numbers k such that A000041(k) == 2 (mod 4).

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A121062 Partition numbers mod 4.

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A237279 Numbers k such that A000041(k) == 1 (mod 4).

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A237277 Numbers k such that A000041(k) == 2 (mod 3).

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