A386257 Numbers k such that k + A067666(k) is a square.

1, 15, 80, 192, 1472, 1482, 1512, 1539, 1938, 2090, 2197, 2370, 2805, 3045, 4095, 4356, 4557, 5796, 5978, 6018, 6156, 7130, 7920, 11445, 12125, 12852, 13578, 13800, 15435, 20405, 26562, 29375, 29592, 30996, 31141, 31682, 32205, 42975, 45733, 46060, 49218, 50652, 51645, 51834, 52767, 54272, 55272

Offset: 1

Author

Will Gosnell and Robert Israel, Jul 16 2025

Keywords

• nonn

Comments

Numbers k such that the sum of k and the squares of its prime factors with multiplicity is a square.

The only term that is a semiprime is 15.

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.

Examples

			a(4) = 192 is a term because 192 = 2^6 * 3 and 192 + 6 * 2^2 + 3^2 = 225 = 15^2 is a square.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A067666, A386304.

Programs

Maple

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

Mathematica

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 *)

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