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A065428 Numbers k such that no x^2 mod k is prime.

%I A065428 #48 Jul 16 2021 13:15:18
%S A065428 1,2,3,4,5,8,12,15,16,24,28,40,48,56,60,72,88,112,120,168,232,240,280,
%T A065428 312,408,520,760,840,1320,1848
%N A065428 Numbers k such that no x^2 mod k is prime.
%C A065428 All numbers in this sequence except 56 are idoneal (A000926) - _Joerg Arndt_, Jul 13 2005
%C A065428 No more terms < 10^6. - _T. D. Noe_, Aug 10 2007
%C A065428 No more terms < 10^11. - _Charles R Greathouse IV_, Dec 15 2008
%C A065428 Numbers x such that all x^3 mod k are nonprimes are 1, 2, 7, 9, 63, and apparently no more.
%H A065428 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 784
%t A065428 t={}; Do[s=Union[Mod[Range[n]^2,n]]; If[Select[s,PrimeQ]=={}, AppendTo[t,n]], {n,1000}]; t  (* _T. D. Noe_, Aug 10 2007 *)
%t A065428 nx2pQ[n_]:=Module[{m=PowerMod[Range[3n],2,n]},Count[ FindTransientRepeat[ m,2][[2]], _?PrimeQ]==0]; Select[Range[2000],nx2pQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 11 2019 *)
%o A065428 (PARI) for(n=1, 10^9, q=1; for(x=1, n-1, if(isprime(lift(Mod(x,n)^2)), q=0; break())); if(q, print1(n, ", "))); \\ edited, _Joerg Arndt_, Jan 28 2015
%o A065428 (Haskell)
%o A065428 a065428 n = a065428_list !! (n-1)
%o A065428 a065428_list = filter f [1..] where
%o A065428    f x = all (== 0) $
%o A065428          map (a010051' . (`mod` x) . a000290) [a000196 x .. x-1]
%o A065428 -- _Reinhard Zumkeller_, Aug 01 2012, Aug 15 2011
%o A065428 (Python)
%o A065428 from sympy import isprime
%o A065428 def ok(n): return not any(isprime((x**2)%n) for x in range(2, n))
%o A065428 print(list(filter(ok, range(1, 2000)))) # _Michael S. Branicky_, May 08 2021
%Y A065428 Cf. A179402 (x^4 mod n).
%Y A065428 Cf. A010051, A000196, A000290.
%Y A065428 Cf. A214583 (n such that for all k with gcd(n, k) = 1 and n > k^2, n - k^2 is prime).
%K A065428 nonn,nice,hard,more
%O A065428 1,2
%A A065428 _Joerg Arndt_, Nov 16 2001