A254328 - OEIS

%I #28 Feb 17 2019 20:47:07

%S 1,2,3,4,5,8,12,16

%N Numbers k such that all x^2 mod k are squares (including 0 and 1).

%C Are there any more terms > 16?

%C There are no more terms less than 10^12. Probably the sequence is finite. - _Charles R Greathouse IV_, Jan 29 2015

%C This is a subsequence of A303704, so it is full. - _Jianing Song_, Feb 14 2019

%e Terms k <= 16 and the squares mod k:

%e 1: [0]

%e 2: [0, 1]

%e 3: [0, 1, 1]

%e 4: [0, 1, 0, 1]

%e 5: [0, 1, 4, 4, 1]

%e 8: [0, 1, 4, 1, 0, 1, 4, 1]

%e 12: [0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1]

%e 16: [0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1]

%e k = 10 is not a term: in the list of squares mod 10, [0, 1, 4, 9, 6, 5, 6, 9, 4, 1], the numbers 5 and 6 are not squares.

%t f[n_] := Mod[Range[n]^2, n]; Select[Range@ 10000, AllTrue[f@ #, IntegerQ[Sqrt[#]] &] &] (* AllTrue function introduced in version 10; _Michael De Vlieger_, Jan 29 2015 *)

%o (PARI) isok(n)=for(k=2,n-1,if(!issquare(lift(Mod(k,n)^2)),return(0)));return(1);

%o for(n=1,10^9,if(isok(n),print1(n,", ")));

%o (PARI) is(n)=for(k=sqrtint(n)+1,n\2, if(!issquare(k^2%n), return(0))); 1

%o for(m=10,10^6,for(k=0,sqrtint(2*m),if(is(t=m^2-k^2),print(t))))

%o \\ _Charles R Greathouse IV_, Jan 29 2015

%Y Cf. A065428, A254329, A096008, A303704.

%K nonn,fini,full

%O 1,2

%A _Joerg Arndt_, Jan 28 2015

%E Keywords fini and full added by _Jianing Song_, Feb 14 2019