A237278 -id:A237278 - cp's OEIS frontend

A225324 Partition numbers of the form 4k.

56, 176, 792, 2436, 5604, 451276, 715220, 831820, 1300156, 2323520, 4087968, 7089500, 8118264, 12132164, 15796476, 26543660, 92669720, 118114304, 150198136, 190569292, 384276336, 483502844, 541946240, 761002156, 851376628, 1188908248, 1327710076, 1844349560

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008586 and A000041.

			56 is in the sequence because 4*14 = 56 and 56 is a partition number: p(11) = A000041(11) = 56.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000041, A008586, A052001, A072871, A087183, A127544, A216258, A225325, A225326, A225327, A225358, A225360, A225361, A237278.

Select[PartitionsP[Range[300]], Mod[#, 4] == 0 &] (* T. D. Noe, May 05 2013 *)

for(n=9, 1e3, t=numbpart(n); if(t%4, , print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

a(n) = 4*A216258(n). - Omar E. Pol, May 08 2013

a(n) = A000041(A237278(n)). - Amiram Eldar, May 22 2025

a(6)-a(28) from T. D. Noe, May 05 2013

A243935 Numbers m such that 5 divides A000041(m).

Original entry on oeis.org

4, 7, 9, 14, 18, 19, 23, 24, 27, 29, 34, 38, 39, 44, 49, 54, 58, 59, 61, 64, 66, 68, 69, 71, 74, 79, 82, 84, 89, 94, 97, 99, 103, 104, 109, 114, 119, 120, 124, 127, 128, 129, 130, 134, 136, 139, 140, 142, 144, 149, 154, 159, 163, 164, 165, 169, 170, 173, 174

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), this sequence (k=5), A243936 (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..200] | IsZero(NumberOfPartitions(n) mod 5)];

Select[Range[200], Mod[PartitionsP[#], 5] == 0 &]

is(n)=numbpart(n)%5==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..200) if mod(A000041(n),5) == 0]

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 *)

A237281 Numbers k such that A000041(k) == 3 (mod 4).

Original entry on oeis.org

3, 5, 6, 7, 14, 16, 20, 23, 24, 33, 35, 38, 41, 44, 51, 53, 54, 56, 60, 63, 68, 72, 76, 77, 81, 82, 91, 92, 95, 99, 102, 111, 115, 118, 121, 127, 134, 138, 139, 140, 146, 156, 159, 161, 164, 165, 166, 168, 169, 173, 177, 178, 182, 183, 188, 192, 196, 201

Offset: 1

Clark Kimberling, Feb 05 2014

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

			A000041(6) = 11 == 3 (mod 4).

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

Cf. A000041, A237276, A237278, A237279, A237280.

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 *)

A243936 Numbers m such that 7 divides A000041(m).

Original entry on oeis.org

5, 10, 11, 12, 16, 18, 19, 24, 26, 27, 33, 37, 39, 40, 41, 47, 48, 52, 53, 54, 55, 61, 68, 75, 76, 82, 83, 89, 96, 97, 103, 110, 111, 117, 124, 125, 131, 138, 140, 145, 147, 152, 159, 166, 170, 173, 177, 180, 187, 191, 194, 201, 208, 213, 215, 222, 225, 229, 232

Offset: 1

Bruno Berselli, Jun 15 2014

Bruno Berselli, Table of n, a(n) for n = 1..1000

Numbers m such that k divides A000041(m), where k is prime: A001560 (k=2), A083214 (k=3), A243935 (k=5), this sequence (k=7), A027827 (k=11), A071750 (k=13). For k composite: A237278 (k=4), A035700 (k=12).

[n: n in [1..250] | IsZero(NumberOfPartitions(n) mod 7)];

Select[Range[250], Mod[PartitionsP[#], 7] == 0 &]

is(n)=numbpart(n)%7==0 \\ Charles R Greathouse IV, Apr 08 2015

# From Peter Luschny in A000041
@CachedFunction
def A000041(n):
    if n == 0: return 1
    S = 0; J = n-1; k = 2
    while 0 <= J:
        T = A000041(J)
        S = S+T if is_odd(k//2) else S-T
        J -= k if is_odd(k) else k//2
        k += 1
    return S
[n for n in (0..250) if mod(A000041(n),7) == 0]

cp's OEIS Frontend

A225324 Partition numbers of the form 4k.

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A243935 Numbers m such that 5 divides A000041(m).

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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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A237281 Numbers k such that A000041(k) == 3 (mod 4).

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A121062 Partition numbers mod 4.

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Maple

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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).