A178751 -id:A178751 - cp's OEIS frontend

A274003 Primitive terms (not equal twice a smaller term) of A178751: moduli n such that x^y == -1 (mod n) only for x = -1 (mod n).

2, 3, 15, 20, 51, 68, 255, 340, 771, 1028, 3855, 5140, 13107, 17476, 65535, 87380, 196611, 262148, 983055

Offset: 1

M. F. Hasler, Jun 06 2016

See A178751 for further information. In particular, A038192 is the subsequence of odd terms.

is_A274003(n)={is_A178751(n) && (bittest(n,0) || !is_A178751(n\2))}

select( n -> bittest(n,0) || !setsearch(A178751,n\2), A178751) \\ assuming the vector A178751 holds enough terms of that sequence.

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.

Original entry on oeis.org

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

Emmanuel Amiot, Mar 19 2007

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

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

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).

Cf. A008784, A000010.

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

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}]

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

Edited by M. F. Hasler, Jun 06 2016

A274021 Least positive x < n-1 such that x^y == -1 (mod n) for some y > 1, or 0 if no such x exist.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 3, 0, 2, 3, 2, 0, 2, 3, 0, 0, 2, 5, 2, 0, 5, 7, 5, 0, 2, 5, 2, 3, 2, 0, 3, 0, 2, 3, 19, 11, 2, 3, 17, 0, 2, 5, 2, 7, 14, 5, 5, 0, 3, 3, 0, 23, 2, 5, 19, 31, 2, 3, 2, 0, 2, 3, 5, 0, 2, 17, 2, 0, 5, 19, 7, 23, 3, 3, 14, 3, 6, 17, 3, 0, 2, 3, 2, 47, 13, 3, 5, 7, 3, 29, 10

Offset: 1

M. F. Hasler, Jun 07 2016

nonn

Indices of nonzero terms are listed in A126949 (in this sense the present sequence can be seen as characteristic function of A126949), indices of zeros (except for n=1) are given in A178751. Without the restriction x < n-1, one would have a(n) = n-1 instead of the zeros, since (n-1)^3 = (-1)^3 = -1 (mod n) for all n.

Cf. A126949, A178751.

A274021(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))}

cp's OEIS Frontend

A274003 Primitive terms (not equal twice a smaller term) of A178751: moduli n such that x^y == -1 (mod n) only for x = -1 (mod n).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A274021 Least positive x < n-1 such that x^y == -1 (mod n) for some y > 1, or 0 if no such x exist.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI