A085712 Semiprimes n such that lpf(n)^spf(n)+1 is also semiprime, where lpf(n) is larger prime factor of n and spf(n) is smaller prime factor of n.

6, 10, 22, 38, 58, 118, 122, 142, 158, 202, 262, 278, 362, 398, 542, 698, 758, 818, 898, 922, 1042, 1138, 1142, 1262, 1282, 1322, 1478, 1502, 1642, 1762, 1858, 1982, 2062, 2078, 2102, 2138, 2182, 2258, 2302, 2342, 2362, 2722, 2878, 2918, 2978, 2998, 3062

Offset

1,1

Comments

There can never be an odd term! Proof. Any odd semiprime is of the form p*q, with 2 < p <= q. Therefore p^q +1 is even. Also p^q+1 is divisible by p+1 and p+1 is greater than 3 and it is even. Therefore p+1 has at least two divisors and we are still left with the factor (p^q+1)/(p+1). QED

Links

Table of n, a(n) for n=1..47.

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method

Example

38=2*19 is a member because 19^2+1=362=2*181.

Mathematica

PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], { # [[2]]}] & /@ FactorInteger[n]]; a = {}; Do[p = PrimeFactors[n]; If[ Length[p] == 2 && Length[ PrimeFactors[ p[[2]]^p[[1]] + 1]] == 2, AppendTo[a, n]], {n, 1000}]; a

Crossrefs

Cf. A001358.

Sequence in context: A108605 A216049 A063765 * A337190 A179875 A166305

Adjacent sequences: A085709 A085710 A085711 * A085713 A085714 A085715

Keyword

nonn

Author

Jason Earls, Jul 19 2003

Extensions

Edited and extended by Robert G. Wilson v, Jul 19 2003

Status

approved