%I #26 Feb 02 2017 02:36:45

%S 1,3,5,7,13,17,23,29,33,35,37,39,41,43,49,51,53,61,63,67,69,71,73,77,

%T 81,83,85,87,89,91,93,95,99,105,107,111,115,119,121,123,127,139,143,

%U 145,155,157,159,161,165,169,173,177,181,183,185,189,193,195,199

%N Odd numbers with an odd number of partitions.

%C The original definition was: Numbers n such that A066897(n) is an odd number.

%C The sequence A281708(n) = b(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

%H Vincenzo Librandi, <a href="/A067567/b067567.txt">Table of n, a(n) for n = 1..1000</a>

%e 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

%p # We conjecture the following program produces the sequence

%p with(combinat):

%p b := n -> add(k^3*numbpart(k)*numbpart(n-k), k = 1..n):

%p c := n -> 2( b(n)/n - floor(b(n)/n) ):

%p for n from 1 to 400 do

%p if c(n) = 1 then print(n/2) end if

%p end do;

%p # _Peter Bala_, Jan 26 2017

%t Select[Range[1, 200, 2], OddQ[PartitionsP[#]] &] (* _T. D. Noe_, Mar 18 2012 *)

%o (PARI) isok(n) = (n % 2) && (numbpart(n) % 2); \\ _Michel Marcus_, Jan 26 2017

%Y Cf. A000041, A066897, A163096, A163097, A194798, A209920, A281708.

%K easy,nonn

%O 1,2

%A _Naohiro Nomoto_, Jan 30 2002

%E New name and more terms from _Omar E. Pol_, Mar 18 2012