A391005 Semiperfect primes: p is a semiperfect prime <=> p is prime and p + 1 is semiperfect.

5, 11, 17, 19, 23, 29, 41, 47, 53, 59, 71, 79, 83, 89, 101, 103, 107, 113, 131, 137, 139, 149, 167, 173, 179, 191, 197, 199, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 347, 349, 353, 359, 367, 379, 383, 389, 401, 419, 431, 439, 443

Offset

1,1

Comments

Note on terminology: We adopt the shorthand 'semiperfect prime' to denote a prime whose successor is semiperfect. This term is not intended to imply that the prime p itself satisfies the semiperfect property, which is impossible for primes.

Semiperfect primes that can be represented as the sum of some subset of the divisors of p + 1 are A391003.

Links

Table of n, a(n) for n=1..55.

Maple

select(n -> isprime(n) and isA005835(n+1), [seq(2..444)]);

Mathematica

pseudoPerfectQ[n_] := Module[{divs = Most[Divisors[n]]}, MemberQ[Total/@Subsets[ divs, Length[ divs]], n]]; Select[Prime[Range[70]], pseudoPerfectQ[#+1]&] (* James C. McMahon, Nov 26 2025 *)

Crossrefs

Cf. A391003, A005835.

Sequence in context: A317048 A081717 A096264 * A391003 A391004 A239709

Adjacent sequences: A391002 A391003 A391004 * A391006 A391007 A391008

Keyword

nonn

Author

Peter Luschny, Nov 26 2025

Status

approved