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

1,2

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.

Formula

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

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 *)

Prog

(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
(PARI) A016726_vec(nMax)={my(S=[], a=1); vector(nMax, n, S=concat(S, n^2); while(#Set(S%a)<n, a++); 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

Crossrefs

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

Sequence in context: A325185 A285532 A350944 * A360136 A359821 A047396

Adjacent sequences: A016723 A016724 A016725 * A016727 A016728 A016729

Keyword

nonn,nice

Author

bernie(AT)wagnerpa.com (Bernie McCabe)

Status

approved