

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.


0



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
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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 * A179875 A166305 A047651
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



