A127219 -id:A127219 - cp's OEIS frontend

A163098 a(n) = A127219(n)/2.

1, 4, 5, 11, 13, 14, 15, 17, 20, 21, 23, 25, 29, 31, 32, 33, 35, 37, 39, 40, 42, 43, 47, 48, 49, 50, 53, 54, 55, 56, 58, 60, 61, 62, 63, 64, 65, 68, 71, 77, 79, 80, 85, 87, 88, 90, 92, 99, 100, 103, 112, 114, 115, 118, 123, 124, 128, 129, 130, 131, 132, 134, 137, 139, 140

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Cf. A000041, A127219, A154798, A163288.

Contribution from R. J. Mathar, Oct 09 2010: (Start)
A127219 := proc(n) option remember; if n = 1 then 2; else for a from procname(n-1)+2 by 2 do if type(combinat[numbpart](a) ,'even') then return a; fi; end do; fi ; end proc:
A163098 := proc(n) A127219(n)/2 ; end proc;
seq(A163098(n),n=1..100) ; (End)

More terms from R. J. Mathar, Oct 09 2010

A154798 Even partition numbers of even numbers.

Original entry on oeis.org

2, 22, 42, 1002, 2436, 3718, 5604, 12310, 37338, 53174, 105558, 204226, 715220, 1300156, 1741630, 2323520, 4087968, 7089500, 12132164, 15796476, 26543660, 34262962, 92669720, 118114304, 150198136, 190569292, 384276336, 483502844

Offset: 1

Omar E. Pol, Jan 26 2009

nonn

Even numbers in A058696.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000041, A005843, A058696, A127219, A154795, A154796, A154797.

aa:= proc(n, i) if n=0 then 1 elif n<0 or i=0 then 0 else aa(n,i):= aa(n, i-1) +aa(n-i, i) fi end: a:= proc(n) local k; if n>1 then a(n-1) fi; for k from `if`(n=1, 0, b(n-1)+2) by 2 while irem(aa(k, k), 2)=1 do od; b(n):= k; aa(k, k) end: seq(a(n), n=1..40); # Alois P. Heinz, Jul 28 2009

Select[Table[PartitionsP[n], {n, 0, 200, 2}], EvenQ] (* Jean-François Alcover, Aug 28 2015 *)

select(x->!(x%2), vector(80, n, numbpart(2*n))) \\ Michel Marcus, Aug 28 2015

More terms from Alois P. Heinz, Jul 28 2009

A163097 Even numbers with an odd number of partitions.

Original entry on oeis.org

0, 4, 6, 12, 14, 16, 18, 20, 24, 32, 36, 38, 44, 48, 52, 54, 56, 60, 68, 72, 76, 82, 88, 90, 92, 102, 104, 114, 118, 132, 134, 138, 140, 144, 146, 148, 150, 152, 156, 162, 164, 166, 168, 172, 178, 182, 186, 188, 190, 192, 194, 196, 202, 204, 208, 210, 212, 214, 216

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Cf. A000041, A005843, A052002, A127219, A154797, A163096, A163099.

Select[2*Range[0,150],OddQ[PartitionsP[#]]&] (* Harvey P. Dale, Nov 13 2013 *)

More terms from Sean A. Irvine, Oct 26 2009

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

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.

A127700 Numbers n such that partition number of n^2 is even.

Original entry on oeis.org

3, 5, 8, 10, 16, 20, 21, 23, 26, 28, 30, 39, 41, 43, 45, 46, 47, 48, 49, 50, 51, 56, 58, 59, 63, 66, 68, 70, 71, 73, 76, 78, 80, 82, 84, 85, 86, 87, 88, 92, 93, 95, 96, 97, 100, 102, 103, 111, 112, 113, 115, 117, 120, 121, 122, 123, 125, 127, 129, 131, 134, 135, 137

Offset: 1

Zak Seidov, Apr 03 2007

nonn

Cf. A000041, A001560, A052001, A052002, A127219.

Select[Range[200],EvenQ[PartitionsP[ #^2]]&]

A000041(n^2) is even.

cp's OEIS Frontend

A163098 a(n) = A127219(n)/2.

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A154798 Even partition numbers of even numbers.

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A163097 Even numbers with an odd number of partitions.

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

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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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A127700 Numbers n such that partition number of n^2 is even.

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