A385664 Numbers y such that there is at least one x with 2 <= x < y, x^2 == y (mod sopfr(y)) and y^2 == x (mod sopfr(x)), where sopfr = A001414.

4, 6, 9, 16, 21, 25, 27, 36, 46, 48, 49, 52, 56, 60, 64, 72, 81, 84, 90, 100, 102, 104, 108, 120, 121, 126, 128, 135, 141, 144, 150, 156, 160, 162, 166, 169, 174, 176, 180, 196, 204, 207, 216, 220, 225, 231, 237, 238, 240, 244, 245, 246, 248, 253, 256, 261, 264, 276, 280, 286, 288, 289, 294, 301

Offset: 1

Author

Will Gosnell and Robert Israel, Aug 03 2025

Keywords

• nonn

Comments

Contains no primes, but all squares of primes (with y = p^2 for p an odd prime, x = p works). Does the sequence contain all squares > 1?

Examples

			a(5) = 21 is a term because with x = 9, we have sopfr(9) = 6, sopfr(21) = 10, 9^2 == 21 == 0 (mod 10) and 21^2 == 9 (mod 16).
		

Links

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

Crossrefs

Cf. A001414.

Programs

Maple

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
filter:= proc(y)
  ormap(x -> x^2 - y mod spf(y) = 0 and y^2 - x mod spf(x) = 0, [$2..y-1])
end proc:
select(filter, [$2 .. 500]);