A093034 Primes of the form 1 + multiple perfect number.

2, 7, 29, 673, 30241, 523777, 2178541, 23569921, 33550337, 66433720321, 137438691329, 30823866178561, 796928461056001, 1802582780370364661761, 9186050031556349952001, 2827987212986831882236723201, 630532357710420079508428362350593, 1928622300236318049928258133164033

Offset: 1

Author

Labos Elemer, May 12 2004

Keywords

• nonn

Comments

From M. F. Hasler and Farideh Firoozbakht, Apr 08 2010: (Start)

Theorem: If p is a term of this sequence and p-1 is a t-perfect number then for each positive integer k, x=p^k is a solution to the equation sigma(phi(x)) = t*(x-1).

Proof: sigma(phi(x)) = sigma(phi(p^k)) = sigma((p-1)*p^(k-1)) = sigma(p-1)*sigma(p^(k-1)) = t*(p-1)*(p^k-1)/(p-1) = t*(p^k-1) = t*(x-1). (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..35

Crossrefs

Cf. A000010 (phi), A000203 (sigma), A007691, A171263, A171264, A171265.

Extensions

a(11)-a(16) from Donovan Johnson, Nov 30 2008

a(17)-a(18) from Amiram Eldar, May 09 2024