A181711 Numbers of the form m*(2^k-1), where m = 2^(k-1)*(2^k-1) is a perfect number (A000396).

18, 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, 4951760152529835082242850816, 6129982163463555428116476125461573244012649752219877376

Offset: 1

Author

Vladimir Shevelev, Nov 07 2010

Keywords

• nonn

Comments

The associated exponents k are in A000043: 2, 3, 5, 7, 13, 17, 19 ,31, 61, ...

One can prove that, if m = 2^(k-1)*(2^k-1) is a perfect number, then m*2^k and m*(2^k-1) are both in A181595. Thus every even term in A000396 is a difference of two terms in A181595.

Examples

			With k=3, m = 2^(k-1)*(2^k - 1) = 2^2*(8 - 1) = 28 is a perfect number (A000396), so m*(2^k - 1) = 28*7 = 196 is in the sequence. - _Michael B. Porter_, Jul 19 2016
		

Crossrefs

Cf. A000043, A000396, A181595, A181596, A181701, A181710.

Formula

If odd perfect numbers do not exist, then a(n) = A181710(n) - A000396(n).

a(n) = A019279(n)*(A000668(n))^2 if there are no odd superperfect numbers. - César Aguilera, Jun 13 2017

Extensions

Definition condensed by R. J. Mathar, Dec 05 2010