A135241 Numbers k such that sigma(sigma(k)) = 2*phi(k).

13, 43, 109, 151, 883, 2143, 116581, 388537, 1711663, 2498227, 4004107, 5550331, 12641137, 13617361, 18591967, 20755393, 22998397, 26838523, 29308291, 34564351, 36300841, 44829073, 82368469, 149460841, 184988197, 238225003, 252757891, 340428853

Offset: 1

Author

Farideh Firoozbakht, Dec 30 2007

Keywords

• nonn

Comments

If p = 2^k + 3 and both numbers p & q = (1/2)*(p^2 - 3p - 2) are primes then q is in the sequence, because sigma(sigma(q)) = sigma(q+1) = sigma((1/2)*(p-3)*p) = sigma(2^(n-1)*p) = (2^n-1)*(p+1) = (p-4)*(p+1) = p^2 - 3p - 4 = 2q - 2 = 2*phi(q). 13, 43, 151 & 2143 are such terms corresponding to n = 2, 3, 4 & 6.

Examples

			sigma(sigma(36300841)) = sigma(36313684) = 72576000 = 2*36288000 = 2*phi(36300841) so 36300841 is in the sequence.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..54 (terms < 10^10)

Programs

Mathematica

lst = {}; fQ[n_] := DivisorSigma[1, DivisorSigma[1, n]] == 2 EulerPhi@n; Do[ If[ fQ@n, AppendTo[lst, n]; Print@n], {n, 252000000}] (* Robert G. Wilson v, Jan 01 2008 *)

PARI

is(n) = sigma(sigma(n))==2*eulerphi(n) \\ Felix Fröhlich, May 18 2019

Extensions

More terms from Robert G. Wilson v, Jan 01 2008