A068907 -id:A068907 - cp's OEIS frontend

A020919 Partition numbers mod 11.

1, 1, 2, 3, 5, 7, 0, 4, 0, 8, 9, 1, 0, 2, 3, 0, 0, 0, 0, 6, 0, 0, 1, 1, 2, 0, 5, 7, 0, 0, 5, 0, 0, 1, 1, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 1, 2, 3, 5, 0, 0, 0, 0, 8, 0, 1, 0, 2, 0, 0, 7, 0, 0, 6, 0, 9, 1, 1, 2, 3, 5, 7, 0, 4, 0, 0, 9, 1, 0, 2, 3, 0, 0, 0, 0, 0, 8, 0, 1, 1, 2, 3, 5, 7, 0, 4, 0, 8, 0, 1, 1, 2, 3, 5, 0

Offset: 0

David W. Wilson

Zeros are uncommonly dense at beginning of sequence, no other modulus exhibits this behavior.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000041, A040051, A068907, A068908, A020919.

Mod[PartitionsP[Range[0,120]],11] (* Harvey P. Dale, May 14 2023 *)

a(n) = numbpart(n) % 11; \\ Michel Marcus, Jul 14 2022

a(n) = A000041(n) mod 11. - Sean A. Irvine, May 04 2019

A087183 Partition numbers of the form 3*k.

Original entry on oeis.org

3, 15, 30, 42, 135, 231, 297, 627, 792, 1002, 1575, 2436, 5604, 8349, 10143, 14883, 31185, 37338, 44583, 63261, 105558, 147273, 239943, 281589, 329931, 614154, 1121505, 1505499, 3087735, 4087968, 4697205, 8118264, 15796476, 44108109

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

The numbers m such that 3 divides A000041(m) are given in A083214. Klarreich writes: no one has proved whether there are infinitely many partition numbers divisible by 3, although it is known that there are infinitely many partition numbers divisible by 2. - Jonathan Vos Post, Jul 31 2008

Intersection of A008585 and A000041. - Reinhard Zumkeller, Nov 03 2009

Erica Klarreich, Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes, Science News, Jun 18 2005.

Paul Tek, Table of n, a(n) for n = 1..10000
Erica Klarreich, Pieces of numbers: a proof brings closure to a dramatic tale of partitions and primes, Science News, Jun 18 2005.
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A008585, A068907, A052001, A052003, A079978, A083214, A087180, A087184, A087185, A167392, A213365.

Select[PartitionsP@Range[120], Divisible[#, 3] &] (* Vladimir Reshetnikov, Nov 05 2015 *)

for(n=9, 1e3, t=numbpart(n); if(t%3, , print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

A079978(a(n))*A167392(a(n)) = 1. - Reinhard Zumkeller, Nov 03 2009
a(n) = 3*A213365(n). - Omar E. Pol, May 08 2013
a(n) = A000041(A083214(n)). - Amiram Eldar, May 22 2025

A068908 Number of partitions of n modulo 5.

Original entry on oeis.org

1, 1, 2, 3, 0, 2, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 1, 2, 0, 0, 2, 2, 2, 0, 0, 3, 1, 0, 3, 0, 4, 2, 4, 3, 0, 3, 2, 2, 0, 0, 3, 3, 4, 1, 0, 4, 3, 4, 3, 0, 1, 3, 4, 1, 0, 1, 3, 4, 0, 0, 2, 0, 1, 4, 0, 3, 0, 4, 0, 0, 3, 0, 3, 4, 0, 4, 1, 3, 4, 0, 1, 2, 0, 4, 0, 2, 2, 3, 4, 0, 3, 4, 2, 2, 0, 4, 4, 0, 1, 0, 2, 1, 4, 0, 0

Offset: 0

Henry Bottomley, Mar 05 2002

nonn

Of the partitions of numbers from 1 to 100000: 36256 are 0, 15758 are 1, 16133 are 2, 16028 are 3 and 15825 are 4 modulo 5, largely because the number of partitions of 5m+4 is always a multiple of 5.

Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Cf. A000041, A010874, A068906.

Cf. A040051, A068907, A068909, A020919.

Mod[PartitionsP[Range[0,110]],5] (* Harvey P. Dale, Dec 20 2023 *)

a(n) = numbpart(n) % 5; \\ Michel Marcus, Jul 14 2022

a(n) = A010874(A000041(n)) = A068906(5, n).

a(n) = Pm(n,1) with Pm(n,k) = if kReinhard Zumkeller, Jun 09 2009]

A087180 Number partition numbers <= P(n) of the form 3*k (P = A000041).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 7, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 15, 15, 16, 16, 16, 16, 17, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 28, 28, 28, 29, 29, 30, 31, 31, 31, 31, 32

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A068907, A071754, A087177, A087181, A087182, A087183.

Accumulate[Table[Boole[Mod[PartitionsP[n], 3] == 0], {n, 0, 100}]] (* Amiram Eldar, May 22 2025 *)

a(n) + A087181(n) + A087182(n) = n + 1.

A087181 Number partition numbers <= P(n) of the form 3*k+1 (P = A000041).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 23, 23, 23

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A068907, A071754, A087177, A087180, A087182, A087184.

Accumulate[Table[Boole[Mod[PartitionsP[n], 3] == 1], {n, 0, 100}]] (* Amiram Eldar, May 22 2025 *)

A087180(n) + a(n) + A087182(n) = n + 1.

A087182 Number partition numbers <= P(n) of the form 3*k+2 (P = A000041).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 17, 17, 18, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 24

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A068907, A071754, A087177, A087180, A087181, A087185.

Accumulate[Table[Boole[Mod[PartitionsP[n], 3] == 2], {n, 0, 100}]] (* Amiram Eldar, May 22 2025 *)

A087180(n) + A087181(n) + a(n) = n + 1.

A087184 Partition numbers of the form 3*k+1.

Original entry on oeis.org

1, 1, 7, 22, 385, 490, 1255, 3010, 3718, 12310, 17977, 21637, 75175, 89134, 204226, 386155, 451276, 831820, 1300156, 1741630, 5392783, 6185689, 10619863, 18004327, 20506255, 34262962, 49995925, 64112359, 104651419, 150198136

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A087181, A087183, A087185, A068907, A052001, A052003, A237276.

Select[PartitionsP[Range[0, 100]], Mod[#, 3] == 1 &] (* Amiram Eldar, May 22 2025 *)

a(n) = A000041(A237276(n)). - Amiram Eldar, May 22 2025

A087185 Partition numbers of the form 3*k+2.

Original entry on oeis.org

2, 5, 11, 56, 77, 101, 176, 1958, 4565, 6842, 26015, 53174, 124754, 173525, 526823, 715220, 966467, 2012558, 2323520, 2679689, 3554345, 7089500, 9289091, 12132164, 13848650, 23338469, 26543660, 30167357, 38887673, 56634173

Offset: 1

Reinhard Zumkeller, Aug 23 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A087182, A087183, A087184, A068907, A052001, A052003, A237277.

Select[PartitionsP[Range[200]],Divisible[#-2,3]&] (* Harvey P. Dale, Apr 22 2016 *)

a(n) = A000041(A237277(n)). - Amiram Eldar, May 22 2025

A068906 Square array read by ascending antidiagonals of partitions of k modulo n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Mar 05 2002

0 is disproportionately common modulo 5, 7 and 11, largely because T(5,5m+4)=T(7,7m+5)=T(11,11m+6)=0.

			Rows start 0,0,0,0,0,...; 1,0,1,1,1,...; 1,2,0,2,1,...; 1,2,3,1,3,...; 1,2,3,0,2,1,...; 1,2,3,5,1,5,...; 1,2,3,5,0,...; 1,2,3,5,7,...; etc.

Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Rows 2, 3, 5, 7 and 11 give A040051, A068907, A068908, A068909, A020919.

T(n, k) =A051127(n, A000041(k))

A068909 Number of partitions of n modulo 7.

Original entry on oeis.org

1, 1, 2, 3, 5, 0, 4, 1, 1, 2, 0, 0, 0, 3, 2, 1, 0, 3, 0, 0, 4, 1, 1, 2, 0, 5, 0, 0, 1, 1, 4, 3, 5, 0, 4, 1, 1, 0, 3, 0, 0, 0, 2, 2, 2, 3, 5, 0, 0, 2, 1, 4, 0, 0, 0, 0, 3, 2, 2, 3, 5, 0, 4, 2, 2, 2, 3, 5, 0, 4, 3, 2, 4, 6, 5, 0, 0, 2, 2, 4, 3, 5, 0, 0, 3, 3, 6, 6, 3, 0, 1, 3, 3, 4, 3, 5, 0, 0, 4, 3, 4, 6, 5, 0, 1

Offset: 0

Henry Bottomley, Mar 05 2002

nonn

Of the partitions of numbers from 1 to 100000: 27193 are 0, 12078 are 1, 12203 are 2, 12260 are 3, 12231 are 4, 12003 are 5 and 12032 are 6 modulo 7, largely because the number of partitions of 7m+5 is always a multiple of 7.

Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Cf. A000041, A010876, A068906.

Cf. A040051, A068907, A068908, A020919.

Table[Mod[PartitionsP[n],7],{n,0,110}] (* Harvey P. Dale, Feb 17 2018 *)

a(n) = numbpart(n) % 7; \\ Michel Marcus, Jul 14 2022

a(n) = A010876(A000041(n)) = A068906(7, n).

a(n) = Pm(n,1) with Pm(n,k) = if kReinhard Zumkeller, Jun 09 2009]

cp's OEIS Frontend

A020919 Partition numbers mod 11.

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A087183 Partition numbers of the form 3*k.

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A068908 Number of partitions of n modulo 5.

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A087180 Number partition numbers <= P(n) of the form 3*k (P = A000041).

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A087181 Number partition numbers <= P(n) of the form 3*k+1 (P = A000041).

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A087182 Number partition numbers <= P(n) of the form 3*k+2 (P = A000041).

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A087184 Partition numbers of the form 3*k+1.

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A087185 Partition numbers of the form 3*k+2.

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