A052001 -id:A052001 - cp's OEIS frontend

A225360 Partition numbers of the form 9k.

135, 297, 792, 1575, 10143, 31185, 63261, 329931, 15796476, 44108109, 4835271870, 7346629512, 12292341831, 18440293320, 107438159466, 129913904637, 156919475295, 250438925115, 1527273599625, 3345365983698, 3646072432125, 7206841706490

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008591 and A000041.

			135 is in the sequence because 9*15 = 135 and 135 is a partition number: p(14) = A000041(14) = 135.

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

Cf. A000041, A008591, A052001, A072871, A087183, A225324, A225325, A225326, A225327, A225358, A127544, A225361.

Select[PartitionsP[Range[300]], Mod[#, 9] == 0 &]

a(n) = 9*A222179(n).

A225361 Partition numbers of the form 11k.

Original entry on oeis.org

11, 22, 77, 176, 231, 297, 385, 627, 792, 1958, 3718, 4565, 6842, 8349, 14883, 21637, 26015, 31185, 44583, 53174, 63261, 173525, 204226, 239943, 281589, 386155, 526823, 715220, 831820, 1121505, 1300156, 1741630, 5392783, 7089500, 8118264, 12132164, 18004327

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008593 and A000041.

			22 is in the sequence because 11*2 = 22 and 22 is a partition number: p(8) = A000041(8) = 22.

Cf. A000041, A008593, A052001, A072871, A087183, A225324, A225325, A225326, A225327, A225358, A225360, A127544.

Select[PartitionsP[Range[300]], Mod[#, 11] == 0 &]

a(n) = 11*A225323(n).

A194798 Numbers n having the same parity as the number of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 13, 17, 22, 23, 26, 28, 29, 30, 33, 34, 35, 37, 39, 40, 41, 42, 43, 46, 49, 50, 51, 53, 58, 61, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 77, 78, 80, 81, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 96, 98, 99, 100, 105, 106, 107, 108, 110, 111

Offset: 1

Omar E. Pol, Jan 29 2012

Odd positive integers with an odd number of partitions and even positive integers with an even number of partitions. - Omar E. Pol, Mar 17 2012

Union of A067567 and A127219. Note that the union of A163096 and A163097 gives A209920 and the union of A209920 and this sequence gives A001477. - Omar E. Pol, Mar 22 2012

			10 is in the sequence because the number of partitions of 10 is equal to 42 and both 10 and 42 have the same parity.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659, A209920.

with(combinat):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while irem(k+numbpart(k), 2)=1 do od; k
    end:
seq(a(n), n=1..80); # Alois P. Heinz, Mar 16 2012

Select[Range[200], Mod[PartitionsP[#] - #, 2] == 0 &] (* T. D. Noe, Mar 16 2012 *)

More terms from Alois P. Heinz, Mar 16 2012

A087177 Number of even partition numbers <= P(n), where P=A000041.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 9, 9, 9, 10, 11, 12, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 20, 21, 21, 21, 22, 22, 22, 22, 22, 23, 23, 24, 25, 26, 26, 26, 27, 27, 28, 29, 30, 30, 30, 30, 31, 31, 31, 31, 32, 33, 33

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Eric Weisstein's World of Mathematics, Partition Function
Eric Weisstein's World of Mathematics, Partition Function P Congruences

Cf. A000041, A052001, A040051, A071754.

a(n) = sum(k=1, n, !(numbpart(k) % 2)); \\ Michel Marcus, Feb 24 2023

a(n) = n + 1 - A071754(n).

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

A209920 Numbers n having distinct parity as the number of partitions of n.

Original entry on oeis.org

0, 4, 6, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 24, 25, 27, 31, 32, 36, 38, 44, 45, 47, 48, 52, 54, 55, 56, 57, 59, 60, 65, 68, 72, 75, 76, 79, 82, 88, 90, 92, 97, 101, 102, 103, 104, 109, 113, 114, 117, 118, 125, 129, 131, 132, 133, 134, 135, 137, 138, 140

Offset: 1

Omar E. Pol, Mar 16 2012

nonn

Odd positive integers with an even number of partitions and nonnegative even integers with an odd number of partitions. Union of A163097 and A163096. Note that the union of A067567 and A127219 gives A194798 and the union of A194798 and this sequence gives A001477.

			4 is in the sequence because the number of partitions of 4 is equal to 5 and the parity of 4 is distinct to the parity of 5 because 4 is even and 5 is odd.
9 is in the sequence because the number of partitions of 9 is equal to 30 and the parity of 9 is distinct to the parity of 30 because 9 is odd and 30 is even.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Complement of A194798.

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659.

A209658 Partition numbers p(n) having the same parity as n.

Original entry on oeis.org

1, 2, 3, 7, 15, 22, 42, 101, 297, 1002, 1255, 2436, 3718, 4565, 5604, 10143, 12310, 14883, 21637, 31185, 37338, 44583, 53174, 63261, 105558, 173525, 204226, 239943, 329931, 715220, 1121505, 1300156, 1505499, 1741630, 2323520, 2679689, 3554345

Offset: 1

Omar E. Pol, Mar 22 2012

nonn

Union of A154795 and A154798. The union of A209659 and this sequence gives A000041.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A194798, A209659, A209920.

A209659 Partition numbers p(n) having opposite parity of n.

Original entry on oeis.org

1, 5, 11, 30, 56, 77, 135, 176, 231, 385, 490, 627, 792, 1575, 1958, 3010, 6842, 8349, 17977, 26015, 75175, 89134, 124754, 147273, 281589, 386155, 451276, 526823, 614154, 831820, 966467, 2012558, 3087735, 5392783, 8118264, 9289091, 13848650

Offset: 1

Omar E. Pol, Mar 22 2012

nonn

Union of A154797 and A154796. The union of this sequence and A209658 gives A000041.

K. Ono, Parity of the partition function, Electronic Research Announcements of AMS, Vol. 1, 1995, pp. 35-42; MR 96d:11108

Cf. A000041, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A194798, A209658, A209920.

A193830 Even partition numbers of prime numbers.

Original entry on oeis.org

2, 56, 490, 6842, 124754, 831820, 13848650, 133230930, 214481126, 271248950, 541946240, 851376628, 5964539504, 11097645016, 37027355200, 45060624582, 142798995930, 207890420102, 625846753120, 1820701100652, 3068829878530, 37561133582570, 114540884553038

Offset: 1

Omar E. Pol, Aug 06 2011

nonn

			The even number 56 is in the sequence as the partition number of the prime number 11.

Even numbers in A058698.

Cf. A000040, A000041, A052001, A154796, A154798, A163997, A193831.

Select[PartitionsP[Prime[Range[200]]],EvenQ] (* Harvey P. Dale, Jun 20 2015 *)

cp's OEIS Frontend

A225360 Partition numbers of the form 9k.

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A225361 Partition numbers of the form 11k.

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A194798 Numbers n having the same parity as the number of partitions of n.

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A087177 Number of even partition numbers <= P(n), where P=A000041.

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PARI

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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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Mathematica

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A209920 Numbers n having distinct parity as the number of partitions of n.

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A209658 Partition numbers p(n) having the same parity as n.

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A209659 Partition numbers p(n) having opposite parity of n.

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