A105331 Numbers of the form 2^n*(2^(n+1)+2n+1) where 2^(n+1)+2n+1 is prime.

3, 14, 52, 184, 656, 34688, 2118656, 134438912, 537346048, 9007202811510784, 2417851639318318791262208, 633825300114170432793740312576, 2535301200456572518883997515776

Offset: 1

Author

Farideh Firoozbakht, Apr 28 2005

Keywords

• nice
• nonn

Comments

This sequence is a subsequence of A083874 (see A083874 & A105330).

This is because these numbers satisfy tau(n) + sigma(n) = 2n when n = 2^k * p with p is prime; for instance tau(14) + sigma(14) = 4 + 24 = 28 = 2 x 14. [See References.] - Bernard Schott, Apr 07 2017

Examples

			9007202811510784 is in the sequence because 9007202811510784 = 2^26*(2^27 + 2*26 + 1) and 2^27 + 2*26 + 1 is prime.
		

References

• J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Problème 723, page 93.

Crossrefs

Cf. A007503, A083874, A105330.

Programs

Mathematica

Do[If[PrimeQ[2^(m + 1) + 2m + 1], Print[2^m(2^(m + 1) + 2m + 1)]], {m, 0, 110}]
2^# (2^(#+1)+2#+1)&/@Select[Range[0,100],PrimeQ[2^(#+1)+2#+1]&] (* Harvey P. Dale, Nov 13 2012 *)

Formula

a(n) = 2^A105330(n)*(2^(A105330(n)+1) + 2*A105330(n) + 1). - Bernard Schott, Apr 07 2017