A108814 Numbers k such that k^4 + 4 is semiprime.

3, 5, 15, 25, 55, 125, 205, 385, 465, 635, 645, 715, 1095, 1145, 1175, 1245, 1275, 1315, 1375, 1565, 1615, 1675, 1685, 1965, 2055, 2085, 2095, 2405, 2455, 2535, 2665, 2835, 2925, 3135, 3305, 3535, 3755, 3775, 4025, 4155, 4175, 4365, 4605, 4615, 4735, 4785

Offset: 1

Author

Jason Earls, Jul 10 2005

Keywords

• nonn

Comments

Except for the first, all the terms above generate brilliant numbers.

Numbers n such that n - 1 + i and n + 1 + i are (twin) Gaussian primes, see Shanks. - Charles R Greathouse IV, Apr 20 2011

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Daniel Shanks, A Note on Gaussian Twin Primes, Mathematics of Computation 14:70 (1960), pp. 201-203.

Crossrefs

Cf. A001358, A057781, A096012.

Programs

Magma

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [ n: n in [1..5000] | IsSemiprime(n^4+4)]; // Vincenzo Librandi, Apr 20 2011

Mathematica

Select[Range[5000],PrimeOmega[#^4+4]==2&] (* Harvey P. Dale, Sep 07 2017 *)

PARI

forstep(n=1,1e5,2,if(isprime(n^2-2*n+2) && isprime(n^2+2*n+2), print1(n", "))) \\ Charles R Greathouse IV, Apr 20 2011

Formula

a(k) = A096012(k) + 1. (Because n^4+4 = ((n-1)^2+1)((n+1)^2+1).) - Jeppe Stig Nielsen, Feb 26 2016