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

%I A386991 #13 Aug 15 2025 02:28:49
%S A386991 1,8,15,35,112,143,323,899,1763,3599,5183,10403,11663,19043,22499,
%T A386991 32399,36863,39203,51983,57599,72899,79523,97343,121103,176399,186623,
%U A386991 213443,272483,324899,359999,381923,412163,435599,656099,675683,685583,736163
%N A386991 Numbers k such that k^2 + sopfr(k)^2 is a square, where sopfr = A001414.
%C A386991 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.
%C A386991 Are 1, 8 and 112 the only terms not in A037074?
%H A386991 Robert Israel, <a href="/A386991/b386991.txt">Table of n, a(n) for n = 1..300</a>
%e A386991 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.
%p A386991 sopfr:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
%p A386991 filter:= t -> issqr(t^2 + sopfr(t)^2):
%p A386991 select(filter, [$1..10^5]);
%t A386991 Sopfr[1]=0;Sopfr[n_]:= Plus @@ Times @@@ FactorInteger@ n;Select[Range[500000],IntegerQ[Sqrt[#^2+Sopfr[#]^2]]&] (* _James C. McMahon_, Aug 14 2025 *)
%Y A386991 Cf. A001414, A386246. Includes A037074.
%K A386991 nonn
%O A386991 1,2
%A A386991 _Robert Israel_, Aug 12 2025