A126949 Moduli n for which -1 is a (nontrivial) power residue for some power greater than 2, i.e., m^k == -1 (mod n) for some k > 1 and some 1 < m < n-1.

5, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset

1,1

Comments

The complement of A178751 (within integers > 1). - M. F. Hasler, Jun 06 2016

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Emmanuel Amiot, Autosimilar Melodies, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.

Emmanuel Amiot, Mélodies autosimilaires (Self-Replicating Melodies) (in French).

Example

19 is in the sequence because -1 == 10^9 (mod 19).

Mathematica

ord[x_, n_] := Module[{k = 1}, While[k <= EulerPhi[n]/2 && PowerMod[x, k, n] != n - 1, k++ ]; If[PowerMod[x, k, n] == n - 1, k, infinity]] iGeneralise[n_] := Module[{candidats = Range[n - 2]}, candidats = Select[candidats, (GCD[n, # ] == 1) &]; Select[candidats, (ord[ #, n] < n) &] ] sol = {}; Do[If[iGeneralise[n] != {}, AppendTo[sol, n]], {n, 2, 100}]

Prog

(Haskell)
a126949 n = a126949_list !! (n-1)
a126949_list = filter h [1..] where
   h m = not $ null [(x, e) | x <- [2 .. m - 2], gcd x m == 1,
                              e <- [2 .. a000010 m `div` 2],
                              x ^ e `mod` m == m - 1]
-- Reinhard Zumkeller, May 23 2013
(PARI) is_A126949(n)={for(x=2, n-2, gcd(x, n)>1&&next; my(t=Mod(x, n)); while(abs(centerlift(t))>1, t*=x); t==-1&&return(x))} \\ (Based on code for A178751 by Ch. Greathouse.) - M. F. Hasler, Jun 07 2016

Crossrefs

Cf. A008784, A000010.

Sequence in context: A194394 A160811 A116451 * A079696 A275718 A129270

Adjacent sequences: A126946 A126947 A126948 * A126950 A126951 A126952

Keyword

nonn,hear

Author

Emmanuel Amiot, Mar 19 2007

Extensions

Edited by M. F. Hasler, Jun 06 2016

Status

approved