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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.

%I A385664 #10 Aug 10 2025 22:05:38
%S A385664 4,6,9,16,21,25,27,36,46,48,49,52,56,60,64,72,81,84,90,100,102,104,
%T A385664 108,120,121,126,128,135,141,144,150,156,160,162,166,169,174,176,180,
%U A385664 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
%N 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.
%C A385664 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?
%H A385664 Robert Israel, <a href="/A385664/b385664.txt">Table of n, a(n) for n = 1..10000</a>
%e A385664 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).
%p A385664 spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
%p A385664 filter:= proc(y)
%p A385664   ormap(x -> x^2 - y mod spf(y) = 0 and y^2 - x mod spf(x) = 0, [$2..y-1])
%p A385664 end proc:
%p A385664 select(filter, [$2 .. 500]);
%Y A385664 Cf. A001414.
%K A385664 nonn
%O A385664 1,1
%A A385664 _Will Gosnell_ and _Robert Israel_, Aug 03 2025