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

Author

Omar E. Pol, Jan 29 2012

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A000041, A040051, A052001, A052003, A067567, A127219, A154795-A154798, A163096, A163097, A163998, A194807, A209658, A209659, A209920.

Programs

Maple

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

Mathematica

Select[Range[200], Mod[PartitionsP[#] - #, 2] == 0 &] (* T. D. Noe, Mar 16 2012 *)

Extensions

More terms from Alois P. Heinz, Mar 16 2012