A052003 -id:A052003 - cp's OEIS frontend

A052001 Even partition numbers.

2, 22, 30, 42, 56, 176, 490, 792, 1002, 1958, 2436, 3010, 3718, 5604, 6842, 12310, 37338, 53174, 89134, 105558, 124754, 204226, 451276, 614154, 715220, 831820, 1300156, 1741630, 2012558, 2323520, 4087968, 7089500, 8118264, 12132164

Offset: 1

Patrick De Geest, Nov 15 1999

nonn

Intersection of A005843 and A000041; A059841(a(n)) * A167392(a(n)) = 1. [Reinhard Zumkeller, Nov 03 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000041, A001560, A005843, A052002, A052003, A059841, A167392, A213179.

a052001 n = a052001_list !! (n-1)
a052001_list = filter even a000041_list
-- Reinhard Zumkeller, Nov 03 2015

Select[PartitionsP[Range[100]], EvenQ] (* Jean-François Alcover, Mar 01 2019 *)

for(n=1, 100, if((k=numbpart(n))%2==0, print1(k", "))) \\ Altug Alkan, Nov 02 2015

a(n) = 2*A213179(n). - Omar E. Pol, May 08 2013

Offset corrected by Reinhard Zumkeller, Nov 03 2015

A052002 Numbers with an odd number of partitions.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 12, 13, 14, 16, 17, 18, 20, 23, 24, 29, 32, 33, 35, 36, 37, 38, 39, 41, 43, 44, 48, 49, 51, 52, 53, 54, 56, 60, 61, 63, 67, 68, 69, 71, 72, 73, 76, 77, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 95, 99, 102, 104, 105, 107, 111, 114, 115, 118, 119, 121

Offset: 1

Patrick De Geest, Nov 15 1999

A052003(n) = A000041(a(n+1)). - Reinhard Zumkeller, Nov 03 2015

Also, numbers having an odd number of partitions into distinct odd parts; that is, numbers m such that A000700(m) is odd. For example, 16 is in the list since 16 has 5 partitions into distinct odd parts, namely, 1 + 15, 3 + 13, 5 + 11, 7 + 9 and 1 + 3 + 5 + 7. See Formula section for a proof. - Peter Bala, Jan 22 2017

			From _Gus Wiseman_, Jan 13 2020: (Start)
The partitions of the initial terms are:
  (1)  (3)    (4)     (5)      (6)       (7)
       (21)   (22)    (32)     (33)      (43)
       (111)  (31)    (41)     (42)      (52)
              (211)   (221)    (51)      (61)
              (1111)  (311)    (222)     (322)
                      (2111)   (321)     (331)
                      (11111)  (411)     (421)
                               (2211)    (511)
                               (3111)    (2221)
                               (21111)   (3211)
                               (111111)  (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)
(End)

Clark Kimberling, Table of n, a(n) for n = 1..1000
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).

Cf. A000041, A000700, A001560, A052001, A052003.
The strict version is A001318, with complement A090864.
The version for prime instead of odd numbers is A046063.
The version for squarefree instead of odd numbers is A038630.
The version for set partitions appears to be A032766.
The version for factorizations is A331050.
The version for strict factorizations is A331230.

import Data.List (findIndices)
a052002 n = a052002_list !! (n-1)
a052002_list = findIndices odd a000041_list
-- Reinhard Zumkeller, Nov 03 2015

N:= 1000: # to get all terms <= N
V:= Vector(N+1):
V[1]:= 1:
for i from 1 to (N+1)/2  do
  V[2*i..N+1]:= V[2*i..N+1] + V[1..N-2*i+2] mod 2
od:
select(t -> V[t+1]=1, [$1..N]); # Robert Israel, Jan 22 2017

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[2, k], {k, 0, 1}] (* Clark Kimberling, Jan 05 2014 *)

for(n=0, 200, if(numbpart(n)%2==1, print1(n", "))) \\ Altug Alkan, Nov 02 2015

From Peter Bala, Jan 22 2016: (Start)

Sum_{n>=0} x^a(n) = (1 + x)*(1 + x^3)*(1 + x^5)*... taken modulo 2. Proof: Product_{n>=1} 1 + x^(2*n-1) = Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(2*n-1)) = Product_{n>=1} (1 - x^(2*n))*(1 - x^(4*n-2))/( (1 - x^(2*n)) * (1 - x^(2*n-1)) ) = ( 1 + 2*Sum_{n>=1} (-1)^n*x^(2*n^2) )/(Product_{n>=1} (1 - x^n)) == 1/( Product_{n>=1} (1 - x^n) ) (mod 2). (End)

Offset corrected and b-file adjusted by Reinhard Zumkeller, Nov 03 2015

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

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

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.

A193831 Odd partition numbers of prime numbers.

Original entry on oeis.org

3, 7, 15, 101, 297, 1255, 4565, 21637, 44583, 63261, 329931, 1121505, 2679689, 4697205, 6185689, 23338469, 49995925, 431149389, 3913864295, 13610949895, 80630964769, 362326859895, 749474411781, 2168627105469, 3646072432125, 10085065885767, 27152408925615

Offset: 1

Omar E. Pol, Aug 06 2011

nonn

			The odd number 101 is in the sequence as the partition number of the prime number 13.

Odd numbers in A058698.

Cf. A000040, A000041, A052003, A154795, A163998, A193830.

cp's OEIS Frontend

A052001 Even partition numbers.

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A052002 Numbers with an odd number of partitions.

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A087183 Partition numbers of the form 3*k.

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

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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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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.