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
Examples
19 is in the sequence because -1 == 10^9 (mod 19).
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).
Programs
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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 -
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}] -
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
Extensions
Edited by M. F. Hasler, Jun 06 2016
Comments