A386640 Numbers k such that k + A224787(k) is a square.

1, 225, 270, 1900, 4988, 5656, 6120, 8704, 11180, 16588, 17710, 19228, 24475, 28449, 29458, 32330, 34606, 38088, 39292, 40221, 41181, 42476, 48545, 48640, 53795, 56832, 57288, 64975, 78793, 84925, 86242, 117116, 124135, 128478, 129673, 134044, 136224, 136896, 147149, 150528, 168055, 183141

Offset: 1

Author

Will Gosnell and Robert Israel, Jul 27 2025

Keywords

• nonn

Comments

Numbers k such that the sum of k and the cubes of the prime factors of k, counted with multiplicity, is a square.

Examples

			a(3) = 270 = 2 * 3^3 * 5 is a term because 270 + 2^3 + 3 * 3^3 + 5^3 =  484 = 22^2 is a square.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A224787, A385238, A386246, A386257, A386623.

Programs

Maple

filter:= proc(n) local t;
   issqr(n + add(t[1]^3*t[2],t=ifactors(n)[2]))
end proc:
select(filter, [$1..10^6]);

Mathematica

lim=184000;f[{p_,e_}]:=e*p^3;a224787[k_]:=If[k==1,0,Total[f/@FactorInteger[k]]];q[k_]:=IntegerQ[Sqrt[k+a224787[k]]];Select[Range[lim],q[#]&] (* James C. McMahon, Jul 30 2025 *)