A386304 Numbers k such that k - A067666(k) is a square.

1, 16, 27, 75, 128, 343, 475, 600, 663, 715, 759, 1015, 1845, 2679, 3717, 3933, 4440, 5083, 5325, 5467, 6120, 6210, 6325, 6405, 6859, 7029, 8349, 8541, 8664, 9125, 9960, 12045, 12427, 12535, 13509, 15067, 16677, 18693, 18711, 21783, 22797, 23250, 23560, 24605, 25527, 26496, 26967, 27117, 28557

Offset

1,2

Comments

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

Is there any number other than 1 in both this sequence and A386257?

Contains no semiprimes.

Links

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

Example

a(4) = 75 is a term because 75 = 3 * 5^2 and 75 - 3^2 - 2 * 5^2 = 16 = 4^2 is a square.

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]]]]; Select[Range[29000], Q[#]&] (* James C. McMahon, Jul 23 2025 *)

Prog

(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

Crossrefs

Cf. A067666, A386257.

Sequence in context: A329206 A384537 A280935 * A067650 A123963 A073396

Adjacent sequences: A386301 A386302 A386303 * A386305 A386306 A386307

Keyword

nonn

Author

Will Gosnell and Robert Israel, Jul 17 2025

Status

approved