A163097 -id:A163097 - cp's OEIS frontend

A163099 a(n) = A163097(n)/2.

0, 2, 3, 6, 7, 8, 9, 10, 12, 16, 18, 19, 22, 24, 26, 27, 28, 30, 34, 36, 38, 41, 44, 45, 46, 51, 52, 57, 59, 66, 67, 69, 70, 72, 73, 74, 75, 76, 78, 81, 82, 83, 84, 86, 89, 91, 93, 94, 95, 96, 97, 98, 101, 102, 104, 105, 106, 107, 108

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Cf. A000041, A154797, A163097.

More terms from Max Alekseyev, Aug 23 2013

A067567 Odd numbers with an odd number of partitions.

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 23, 29, 33, 35, 37, 39, 41, 43, 49, 51, 53, 61, 63, 67, 69, 71, 73, 77, 81, 83, 85, 87, 89, 91, 93, 95, 99, 105, 107, 111, 115, 119, 121, 123, 127, 139, 143, 145, 155, 157, 159, 161, 165, 169, 173, 177, 181, 183, 185, 189, 193, 195, 199

Offset: 1

Naohiro Nomoto, Jan 30 2002

The original definition was: Numbers n such that A066897(n) is an odd number.
The sequence defined by b(n) = (n/2)*A281708(n) = Sum_{k=1..n} k^3 * p(k) * p(n-k) of Peter Bala appears to have the property that b(n)/n is a positive integer if n is odd, and b(2*n)/n is a positive integer which is odd iff n is a member of this sequence. - Michael Somos, Jan 28 2017
From Peter Bala, Jan 10 2025: (Start)
We generalize the above conjecture as follows.
Define b_m(n) = Sum_{k = 1..n} k^(2*m+1) * p(k) * p(n-k). Then for m >= 1,
i) for odd n, b_m(n)/n is an integer
ii) b(2*n)/n is an integer, which is odd iff n is a term of this sequence.
Cf. A067589.
We further conjecture that A305123(n) is odd iff n is a term of this sequence. (End)

			7 is in the sequence because the number of partitions of 7 is equal to 15 and both 7 and 15 are odd numbers. - _Omar E. Pol_, Mar 18 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000041, A066897, A067589, A163096, A163097, A194798, A209920, A281708, A305123.

# We conjecture that the following program produces the sequence
with(combinat):
b := n -> add(k^3*numbpart(k)*numbpart(n-k), k = 1..n):
c := n -> 2( b(n)/n - floor(b(n)/n) ):
for n from 1 to 400 do
  if c(n) = 1 then print(n/2) end if
end do;
# Peter Bala, Jan 26 2017

Select[Range[1, 200, 2], OddQ[PartitionsP[#]] &] (* T. D. Noe, Mar 18 2012 *)

isok(n) = (n % 2) && (numbpart(n) % 2); \\ Michel Marcus, Jan 26 2017

New name and more terms from Omar E. Pol, Mar 18 2012

A163096 Odd numbers with an even number of partitions.

Original entry on oeis.org

9, 11, 15, 19, 21, 25, 27, 31, 45, 47, 55, 57, 59, 65, 75, 79, 97, 101, 103, 109, 113, 117, 125, 129, 131, 133, 135, 137, 141, 147, 149, 151, 153, 163, 167, 171, 175, 179, 187, 191, 197, 205, 207, 213, 217, 227, 231, 241, 243, 245, 247, 253, 255, 265, 267, 271

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000041, A001560, A005408, A154796, A163097.

Select[Range[1,500,2],EvenQ[PartitionsP[#]]&] (* Vincenzo Librandi, Mar 19 2012 *)

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.

cp's OEIS Frontend

A163099 a(n) = A163097(n)/2.

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A067567 Odd numbers with an odd number of partitions.

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A163096 Odd numbers with an even 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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Maple

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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