A386991 Numbers k such that k^2 + sopfr(k)^2 is a square, where sopfr = A001414.

1, 8, 15, 35, 112, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163

Offset: 1

Author

Robert Israel, Aug 12 2025

Keywords

• nonn

Comments

Includes A037074 because if k = p*(p+2) where p and p+2 are primes, k^2 + sopfr(k)^2 = p^2*(p+2)^2 + (2*p+2)^2 = (p^2 + 2*p + 2)^2.

Are 1, 8 and 112 the only terms not in A037074?

Examples

			a(3) = 15 is a term because the sum of prime factors of 15 is 3+5 = 8 and 15^2 + 8^2 = 289 = 17^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A001414, A386246. Includes A037074.

Programs

Maple

sopfr:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
filter:= t -> issqr(t^2 + sopfr(t)^2):
select(filter, [$1..10^5]);

Mathematica

Sopfr[1]=0;Sopfr[n_]:= Plus @@ Times @@@ FactorInteger@ n;Select[Range[500000],IntegerQ[Sqrt[#^2+Sopfr[#]^2]]&] (* James C. McMahon, Aug 14 2025 *)