A110082 -id:A110082 - cp's OEIS frontend

A098855 Numbers k such that 4^k + 2^k - 1 is prime.

1, 2, 3, 4, 6, 10, 16, 24, 26, 35, 52, 55, 95, 144, 379, 484, 939, 1284, 1300, 2651, 3644, 3979, 7179, 8304, 14840, 32040, 38906, 47759, 51371, 52484, 54016, 57279

Offset: 1

Pierre CAMI, Oct 11 2004

Or, numbers k such that (2^k+1)*2^k - 1 is prime.

			1300 is in the sequence because 4^1300+2^1300-1 is prime.
4^1+2^1-1 = 5 prime so a(1)=1.
4^2+2^2-1 = 19 prime so a(2)=2.
4^3+2^3-1 = 71 prime so a(3)=3.
4^4+2^4-1 = 271 prime so a(4)=4.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1

Cf. A110079, A110082, A210733.

Do[If[PrimeQ[4^m+2^m-1], Print[m]], {m, 8000}] (* Farideh Firoozbakht, Aug 03 2005 *)

is(n)=ispseudoprime(4^n+2^n-1) \\ Charles R Greathouse IV, Jun 13 2017

A110082(n) = 2^(a(n)-1)*(4^a(n)+2^a(n)-1).

Corrected by Torin Huzil (thuzil(AT)gmail.com), Sep 15 2005
More terms from Pierre CAMI, May 10 2012
a(27), a(29)-a(31) inserted by Michael S. Branicky, Jan 02 2025

A110079 Numbers n such that sigma(n)=2n-2^d(n) where d(n) is number of positive divisors of n.

Original entry on oeis.org

5, 38, 284, 1370, 2168, 26828, 133088, 1515608, 19414448, 23521328, 25812848, 49353008, 82988756, 103575728, 537394688, 558504608, 921747488, 2651596448, 17517611968, 18249863488, 77792665408, 556915822208

Offset: 1

Farideh Firoozbakht, Aug 03 2005

If 4^m+2^m-1 is prime then n=2^(m-1)*(4^m+2^m-1) is in the sequence because 2n-2^d(n)=2^m*(4^m+2^m-1)-2^(m*2)=2^m* (4^m-1)=2^m*(2^m-1)*(2^m+1)=(2^m-1)*(4^m+2^m)=sigma(2^(m-1)) *sigma(4^m+2^m-1)=sigma(2^(m-1)*(4^m+2^m-1))=sigma(n). A110082 gives such terms of this sequence.
a(22) <= 556915822208. a(23) <= 9311639470208. a(24) <= 29297682437888. - Donovan Johnson, Jan 31 2009
a(23) > 6*10^12. - Giovanni Resta, Aug 14 2013

Cf. A110080-3.

Do[If[DivisorSigma[1, n] == 2n - 2^DivisorSigma[0, n], Print[n]], {n, 925000000}]

a(18)-a(21) from Donovan Johnson, Jan 31 2009

a(22) confirmed by Giovanni Resta, Aug 14 2013

cp's OEIS Frontend

A098855 Numbers k such that 4^k + 2^k - 1 is prime.

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