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A385885 Nonprimes k such that k^2 - sopfr(k)^2 is a square, where sopfr = A001414.

%I A385885 #53 Aug 22 2025 17:38:27
%S A385885 1,4,366,1095,51846,258410,982815,10653351
%N A385885 Nonprimes k such that k^2 - sopfr(k)^2 is a square, where sopfr = A001414.
%e A385885 a(4) = 1095 is a term because 1095 = 3 * 5 * 73 and 1095^2 - (3+5+73)^2 = 1192464 = 1092^2.
%p A385885 sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
%p A385885 filter:= t -> not isprime(t) and issqr(t^2 - sopfr(t)^2):
%p A385885 select(filter, [$1..10^8]);
%t A385885 q[k_] := !PrimeQ[k] && IntegerQ[Sqrt[k^2 - (Plus @@ Times @@@ FactorInteger[k])^2]]; Select[Range[10^6], q] (* _Amiram Eldar_, Aug 14 2025 *)
%o A385885 (Python)
%o A385885 from math import isqrt
%o A385885 from sympy import factorint, isprime
%o A385885 def is_square(n): return n >= 0 and isqrt(n)**2 == n
%o A385885 def ok(n): return  not isprime(n) and is_square(n**2-sum(p*e for p,e in factorint(n).items())**2)
%o A385885 print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Aug 14 2025
%o A385885 (PARI) isok(k) = if (!ispseudoprime(k), my(f=factor(k)); issquare(k^2 - (f[, 1]~*f[, 2])^2)); \\ _Michel Marcus_, Aug 21 2025
%Y A385885 Cf. A001414, A386991.
%K A385885 nonn,more,new
%O A385885 1,2
%A A385885 _Robert Israel_, Aug 13 2025
%E A385885 Edited to insert 1 by _Robert Israel_, Aug 22 2025