A067567 Odd numbers with an odd number of partitions.

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

Author

Naohiro Nomoto, Jan 30 2002

Keywords

• easy
• nonn

Comments

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)

Examples

			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
		

Links

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

Crossrefs

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

Programs

Maple

# 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

Mathematica

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

PARI

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

Extensions

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