A067567 - OEIS

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

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OFFSET

1,2

COMMENTS

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

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

LINKS

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

EXAMPLE

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

MAPLE

# We conjecture 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;