A052001 -id:A052001 - cp's OEIS frontend

A052002 Numbers with an odd number of partitions.

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

A001560 Numbers with an even number of partitions.

Original entry on oeis.org

2, 8, 9, 10, 11, 15, 19, 21, 22, 25, 26, 27, 28, 30, 31, 34, 40, 42, 45, 46, 47, 50, 55, 57, 58, 59, 62, 64, 65, 66, 70, 74, 75, 78, 79, 80, 84, 86, 94, 96, 97, 98, 100, 101, 103, 106, 108, 109, 110, 112, 113, 116, 117, 120, 122, 124, 125, 126, 128, 129, 130, 131

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377-378. MR0117213 (22 #7995).
P. A. MacMahon, The parity of p(n), the number of partitions of n, when n <= 1000, J. London Math. Soc., 1 (1926), 225-226.
T. R. Parkin and D. Shanks, On the distribution of parity in the partition function, Math. Comp., 21 (1967), 466-480.

Cf. A052001, A052002, A000041, A243935.

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

is(n)=numbpart(n)%2==0 \\ Charles R Greathouse IV, Apr 08 2015

A213179 Numbers k such that 2*k is a partition number.

Original entry on oeis.org

1, 11, 15, 21, 28, 88, 245, 396, 501, 979, 1218, 1505, 1859, 2802, 3421, 6155, 18669, 26587, 44567, 52779, 62377, 102113, 225638, 307077, 357610, 415910, 650078, 870815, 1006279, 1161760, 2043984, 3544750, 4059132, 6066082, 6924325, 7898238, 13271830

Offset: 1

Omar E. Pol, Feb 27 2013

nonn

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

Cf. A000041, A052001, A213365, A216258, A217725, A217726, A222175, A222178, A222179, A225317, A225323.

Select[PartitionsP[Range@200]/2, IntegerQ] (* Giovanni Resta, May 05 2013 *)

a(n) = A052001(n)/2.

A225324 Partition numbers of the form 4k.

Original entry on oeis.org

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

A127544 Partition numbers (A000041) which are multiples of 10 (A008592).

Original entry on oeis.org

30, 490, 3010, 12310, 715220, 831820, 1741630, 2323520, 7089500, 13848650, 26543660, 92669720, 133230930, 271248950, 541946240, 1844349560, 2841940500, 4351078600, 4835271870, 5371315400, 10015581680, 18440293320, 37027355200

Offset: 1

Zak Seidov, Apr 01 2007

nonn

Intersection of A000041 and A008592.

Partition numbers of the form 10k. - Omar E. Pol, May 08 2013

Cf. A000041, A052001, A008592, A087183.

Select[Table[PartitionsP[n],{n,0,200}],Mod[ #,10]==0&]

a(n) = 10*A225317(n). - Omar E. Pol, May 08 2013

Corrected by Omar E. Pol, May 05 2013

A225325 Partition numbers of the form 5k.

Original entry on oeis.org

5, 15, 30, 135, 385, 490, 1255, 1575, 3010, 4565, 12310, 26015, 31185, 75175, 173525, 386155, 715220, 831820, 1121505, 1741630, 2323520, 3087735, 3554345, 4697205, 7089500, 13848650, 20506255, 26543660, 49995925, 92669720, 133230930, 169229875

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008587 and A000041.

			15 is in the sequence because 5*3 = 15 and 15 is a partition number: p(7) = A000041(7) = 15.

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

Cf. A000041, A008587, A052001, A072871, A087183, A225324, A225326, A225327, A225358, A225360, A127544, A225361

Select[PartitionsP[Range[300]], Mod[#, 5] == 0 &] (* T. D. Noe, May 05 2013 *)

for(n=9, 1e3, t=numbpart(n); if(t%5, , print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

a(n) = 5*A217725(n). - Omar E. Pol, May 08 2013

A225326 Partition numbers of the form 6k.

Original entry on oeis.org

30, 42, 792, 1002, 2436, 5604, 37338, 105558, 614154, 4087968, 8118264, 15796476, 133230930, 384276336, 2841940500, 3163127352, 4835271870, 7346629512, 18440293320, 30388671978, 45060624582, 107438159466, 142798995930, 684957390936, 1820701100652

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008588 and A000041.

			30 is in the sequence because 6*5 = 30 and 30 is a partition number: p(9) = A000041(9) = 30.

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

Cf. A000041, A008588, A052001, A072871, A087183, A225324, A225325, A225327, A225358, A225360, A127544, A225361

Select[PartitionsP[Range[300]], Mod[#, 6] == 0 &] (* T. D. Noe, May 05 2013 *)

for(n=9,1e3,t=numbpart(n);if(t%6,,print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

a(n) = 6*A217726(n). - Omar E. Pol, May 08 2013

a(8)-a(25) from T. D. Noe, May 05 2013

A225327 Partition numbers of the form 7k.

Original entry on oeis.org

7, 42, 56, 77, 231, 385, 490, 1575, 2436, 3010, 10143, 21637, 31185, 37338, 44583, 124754, 147273, 281589, 329931, 386155, 451276, 1121505, 3087735, 8118264, 9289091, 20506255, 23338469, 49995925, 118114304, 133230930, 271248950, 607163746

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008589 and A000041.

			42 is in the sequence because 7*6 = 42 and 42 is a partition number: p(10) = A000041(10) = 42.

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

Cf. A000041, A008589, A052001, A072871, A087183, A225324, A225325, A225326, A225358, A225360, A127544, A225361.

Select[PartitionsP[Range[300]], Mod[#, 7] == 0 &]

a(n) = 7*A222175(n).

A225358 Partition numbers of the form 8k.

Original entry on oeis.org

56, 176, 792, 2323520, 4087968, 8118264, 92669720, 118114304, 150198136, 384276336, 541946240, 1188908248, 1844349560, 2291320912, 3163127352, 4351078600, 5371315400, 5964539504, 7346629512, 10015581680, 11097645016, 16670689208, 18440293320

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008590 and A000041.

			56 is in the sequence because 8*7 = 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, A008590, A052001, A072871, A087183, A225324, A225325, A225326, A225327, A225360, A127544, A225361.

Select[PartitionsP[Range[300]], Mod[#, 8] == 0 &]

a(n) = 8*A222178(n).

cp's OEIS Frontend

A052002 Numbers with an odd number of partitions.

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A087183 Partition numbers of the form 3*k.

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A001560 Numbers with an even number of partitions.

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A213179 Numbers k such that 2*k is a partition number.

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A225324 Partition numbers of the form 4k.

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A127544 Partition numbers (A000041) which are multiples of 10 (A008592).

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A225325 Partition numbers of the form 5k.

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