This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386257 #15 Jul 23 2025 14:36:37
%S A386257 1,15,80,192,1472,1482,1512,1539,1938,2090,2197,2370,2805,3045,4095,
%T A386257 4356,4557,5796,5978,6018,6156,7130,7920,11445,12125,12852,13578,
%U A386257 13800,15435,20405,26562,29375,29592,30996,31141,31682,32205,42975,45733,46060,49218,50652,51645,51834,52767,54272,55272
%N A386257 Numbers k such that k + A067666(k) is a square.
%C A386257 Numbers k such that the sum of k and the squares of its prime factors with multiplicity is a square.
%C A386257 The only term that is a semiprime is 15.
%C A386257 The generalized Bunyakovsky conjecture implies that there are infinitely many pairs of primes (p,q) with 4 * q = 21 * p^2 - 10 * p - 99. For such p and q, 5*p*q is a term.
%H A386257 Robert Israel, <a href="/A386257/b386257.txt">Table of n, a(n) for n = 1..10000</a>
%e A386257 a(4) = 192 is a term because 192 = 2^6 * 3 and 192 + 6 * 2^2 + 3^2 = 225 = 15^2 is a square.
%p A386257 filter:= proc(n) local t;
%p A386257 issqr(n + add(t[1]^2*t[2],t=ifactors(n)[2]))
%p A386257 end proc:
%p A386257 select(filter, [$1..10^5]);
%t A386257 spf[{p_,e_}]:=e*p^2;Q[k_]:=IntegerQ[Sqrt[k+Total[spf/@FactorInteger[k]]]];Join[{1},Select[Range[56000],Q[#]&]] (* _James C. McMahon_, Jul 23 2025 *)
%o A386257 (PARI) isok(k) = my(f=factor(k)); issquare(k + sum(i=1, #f~, f[i, 1]^2*f[i, 2])); \\ _Michel Marcus_, Jul 20 2025
%Y A386257 Cf. A067666, A386304.
%K A386257 nonn
%O A386257 1,2
%A A386257 _Will Gosnell_ and _Robert Israel_, Jul 16 2025