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

%I A387049 #14 Aug 18 2025 06:29:51
%S A387049 6,10,12,14,21,44,46,51,57,65,74,86,90,111,141,155,161,166,210,212,
%T A387049 221,252,254,295,297,300,306,365,371,404,415,447,466,485,504,513,514,
%U A387049 524,545,629,634,640,674,685,720,767,866,910,914,930,985,1020,1035,1062,1124,1135,1157,1189,1197,1214
%N A387049 Numbers k such that both k + sopfr(k) and k^2 + sopfr(k)^2 are prime, where sopfr = A001414.
%C A387049 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.
%H A387049 Robert Israel, <a href="/A387049/b387049.txt">Table of n, a(n) for n = 1..10000</a>
%e A387049 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.
%p A387049 sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
%p A387049 filter:= proc(n) local s; s:= sopfr(n); isprime(n+s) and isprime(n^2 + s^2) end proc:
%p A387049 select(filter, [$1..2000]);
%t A387049 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 *)
%Y A387049 Cf. A001414. Intersection of A050703 and A387048.
%K A387049 nonn,new
%O A387049 1,1
%A A387049 _Robert Israel_, Aug 14 2025