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A067567 Odd numbers with an odd number of partitions.

%I A067567 #36 Jan 15 2025 04:17:45
%S A067567 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 A067567 81,83,85,87,89,91,93,95,99,105,107,111,115,119,121,123,127,139,143,
%U A067567 145,155,157,159,161,165,169,173,177,181,183,185,189,193,195,199
%N A067567 Odd numbers with an odd number of partitions.
%C A067567 The original definition was: Numbers n such that A066897(n) is an odd number.
%C A067567 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
%C A067567 From _Peter Bala_, Jan 10 2025: (Start)
%C A067567 We generalize the above conjecture as follows.
%C A067567 Define b_m(n) = Sum_{k = 1..n} k^(2*m+1) * p(k) * p(n-k). Then for m >= 1,
%C A067567 i) for odd n, b_m(n)/n is an integer
%C A067567 ii) b(2*n)/n is an integer, which is odd iff n is a term of this sequence.
%C A067567 Cf. A067589.
%C A067567 We further conjecture that A305123(n) is odd iff n is a term of this sequence. (End)
%H A067567 Vincenzo Librandi, <a href="/A067567/b067567.txt">Table of n, a(n) for n = 1..1000</a>
%e A067567 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 A067567 # We conjecture that the following program produces the sequence
%p A067567 with(combinat):
%p A067567 b := n -> add(k^3*numbpart(k)*numbpart(n-k), k = 1..n):
%p A067567 c := n -> 2( b(n)/n - floor(b(n)/n) ):
%p A067567 for n from 1 to 400 do
%p A067567   if c(n) = 1 then print(n/2) end if
%p A067567 end do;
%p A067567 # _Peter Bala_, Jan 26 2017
%t A067567 Select[Range[1, 200, 2], OddQ[PartitionsP[#]] &] (* _T. D. Noe_, Mar 18 2012 *)
%o A067567 (PARI) isok(n) = (n % 2) && (numbpart(n) % 2); \\ _Michel Marcus_, Jan 26 2017
%Y A067567 Cf. A000041, A066897, A067589, A163096, A163097, A194798, A209920, A281708, A305123.
%K A067567 easy,nonn
%O A067567 1,2
%A A067567 _Naohiro Nomoto_, Jan 30 2002
%E A067567 New name and more terms from _Omar E. Pol_, Mar 18 2012