A163998 -id:A163998 - cp's OEIS frontend

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

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

A163997 Primes with an even number of partitions.

Original entry on oeis.org

2, 11, 19, 31, 47, 59, 79, 97, 101, 103, 109, 113, 131, 137, 149, 151, 163, 167, 179, 191, 197, 227, 241, 271, 307, 317, 337, 347, 353, 359, 379, 383, 397, 409, 419, 431, 439, 449, 487, 503, 509, 521, 523, 541, 557, 563, 569, 571, 577, 599, 607, 631, 641, 643

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

			11 is in the sequence because the number of partitions of 11 is equal to 56 and 56 is an even number.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A000041, A001560, A163096, A163998.

Select[Prime[Range[200]],EvenQ[PartitionsP[#]]&] (* Harvey P. Dale, Jul 01 2019 *)

More terms from Sean A. Irvine, Oct 18 2009

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.

A163996 Primes with a composite number of partitions.

Original entry on oeis.org

7, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

Omar E. Pol, Aug 09 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000040, A000041, A002808, A038601, A135010, A163997, A163998.

Select[Prime[Range[100]],!PrimeQ[PartitionsP[#]]&] (* Harvey P. Dale, Jun 22 2014 *)

A000040 \ A038601. [From R. J. Mathar, Sep 27 2009]

More terms from R. J. Mathar, Sep 27 2009

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

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

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A163997 Primes with an even number of partitions.

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

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A163996 Primes with a composite number of partitions.

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A193831 Odd partition numbers of prime numbers.

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