A387049 Numbers k such that both k + sopfr(k) and k^2 + sopfr(k)^2 are prime, where sopfr = A001414.

6, 10, 12, 14, 21, 44, 46, 51, 57, 65, 74, 86, 90, 111, 141, 155, 161, 166, 210, 212, 221, 252, 254, 295, 297, 300, 306, 365, 371, 404, 415, 447, 466, 485, 504, 513, 514, 524, 545, 629, 634, 640, 674, 685, 720, 767, 866, 910, 914, 930, 985, 1020, 1035, 1062, 1124, 1135, 1157, 1189, 1197, 1214

Offset: 1

Author

Robert Israel, Aug 14 2025

Keywords

• nonn
• new

Comments

Includes 2*p if p is a prime such that 3*p + 2 and 5 * p^2 + 4 * p + 4 are prime. The Generalized Bunyakowsky Conjecture implies there are infinitely many of these.

Examples

			a(3) = 12 is a term because sopfr(12) = 2*2 + 3 = 7 and both 12 + 7 = 19 and 12^2 + 7^2 = 193 are prime.
		

Links

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

Crossrefs

Cf. A001414. Intersection of A050703 and A387048.

Programs

Maple

sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
filter:= proc(n) local s; s:= sopfr(n); isprime(n+s) and isprime(n^2 + s^2) end proc:
select(filter, [$1..2000]);

Mathematica

q[k_] := Module[{sopfr = Plus @@ Times @@@ FactorInteger[k]}, PrimeQ[k + sopfr] && PrimeQ[k^2 + sopfr^2]]; Select[Range[2, 1214], q] (* Amiram Eldar, Aug 14 2025 *)