A387048 Numbers k such that k^2 + sopfr(k)^2 is prime, where sopfr = A001414.

6, 10, 12, 14, 21, 22, 39, 40, 44, 46, 51, 54, 57, 62, 65, 69, 74, 80, 82, 86, 90, 91, 95, 104, 108, 111, 115, 119, 129, 134, 141, 155, 161, 164, 166, 172, 176, 187, 189, 202, 210, 212, 217, 221, 226, 232, 244, 248, 252, 254, 265, 272, 274, 287, 292, 295, 297, 299, 300, 302, 305, 306, 328, 339

Offset: 1

Author

Robert Israel, Aug 14 2025

Keywords

• nonn
• new

Comments

Includes 2*p where p is a prime such that 5 * p^2 + 4 * p + 4 is prime. The Generalized Bunyakowsky Conjecture implies there are infinitely many of these.

Examples

			a(3) = 12 is a term because 12^2 + sopfr(12)^2 = 144 + (2*2+3)^2 = 193 is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A050703, A386991, A387049.

Programs

Maple

sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
filter:= t -> isprime(t^2 + sopfr(t)^2):
select(filter, [$1..10^3]);

Mathematica

q[k_] := PrimeQ[k^2 + (Plus @@ Times @@@ FactorInteger[k])^2]; Select[Range[2, 340], q] (* Amiram Eldar, Aug 14 2025 *)