A016726 Smallest k such that 1, 4, 9, ..., n^2 are distinct mod k.

1, 2, 6, 9, 10, 13, 14, 17, 19, 22, 22, 26, 26, 29, 31, 34, 34, 37, 38, 41, 43, 46, 46, 53, 53, 53, 58, 58, 58, 61, 62, 67, 67, 71, 71, 73, 74, 79, 79, 82, 82, 86, 86, 89, 94, 94, 94, 97, 101, 101, 103, 106, 106, 109, 113, 113, 118, 118, 118, 122, 122, 127, 127, 131, 131, 134

Offset: 1

Author

bernie(AT)wagnerpa.com (Bernie McCabe)

Keywords

• nonn
• nice

Comments

This is the sequence of discriminators of the squares A000290, in the terminology of Arnold et al. - M. F. Hasler, May 04 2016

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Arnold, L. K.; Benkoski, S. J.; and McCabe, B. J.; The discriminator (a simple application of Bertrand's postulate). Amer. Math. Monthly 92 (1985), 275-277.

Crossrefs

Cf. A001751, A192419 (cubes), A192420 (4th powers), A347693.

Programs

Haskell

a016726 n = a016726_list !! (n-1)
a016726_list = [1,2,6,9] ++ (f 5 $ drop 4 a001751_list) where
   f n qs'@(q:qs) | q < 2*n   = f n qs
                  | otherwise = q : f (n+1) qs'
-- Reinhard Zumkeller, Jun 20 2011

Mathematica

a[n_] := (k = 2n; While[ Not[PrimeQ[k] || PrimeQ[k/2]], k++]; k); a[1]=1; a[2]=2; a[3]=6; a[4]=9; Table[a[n], {n, 1, 66}] (* Jean-François Alcover, Nov 30 2011, after formula *)
sk[n_]:=Module[{k=2n,n2=Range[n]^2},While[Max[Tally[Mod[n2,k]][[All,2]]]> 1,k++];k]; Join[{1,2},Array[sk,70,3]] (* Harvey P. Dale, Oct 16 2016 *)

PARI

A016726_vec(nMax)={my(S=[], a=1); vector(nMax, n, S=concat(S, n^2); while(#Set(S%a)M. F. Hasler, May 04 2016

PARI

A016726(n)=if(n>4,min(nextprime(2*n),2*nextprime(n)),[1,2,6,9][n]) \\ M. F. Hasler, May 04 2016

Formula

For n > 4, a(n) is smallest k >= 2n such that k = p or k = 2p, p a prime.