A126949 -id:A126949 - cp's OEIS frontend

A178751 Numbers k such that in Z/kZ the equation x^y + 1 = 0 has only the trivial solutions with x == -1 (mod k).

2, 3, 4, 6, 8, 12, 15, 16, 20, 24, 30, 32, 40, 48, 51, 60, 64, 68, 80, 96, 102, 120, 128, 136, 160, 192, 204, 240, 255, 256, 272, 320, 340, 384, 408, 480, 510, 512, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1360, 1536, 1542, 1632, 1920, 2040

Offset: 1

Arnaud Vernier, Jun 09 2010, Jun 10 2010

nonn

It appears that odd terms 3, 15, 51, 255, 771, 3855, 13107, 65535, ... are given by A038192. - Michel Marcus, Aug 08 2013

This is the complement of A126949 in the numbers k > 1. (But it could be argued that the sequence should start with k = 1 as initial term.) It appears that for any a(j) in the sequence, 2*a(j) is also in the sequence. The primitive terms (not of the form a(j) = 2*a(m), m < j) are 2, 3, 15, 20, 51, 68, 255, 340, 771, 1028, .... (see A274003). - M. F. Hasler, Jun 06 2016

			In Z/3Z, the only solution to the equation x^y + 1 = 0 is x = 2 and y = 1. Whereas in Z/5Z, the equation has at least one nontrivial solution: 2^2 + 1 = 0.

Arnaud Vernier and Charles R Greathouse IV, Table of n, a(n) for n = 1..189 (first 78 terms from Vernier)

Cf. A038192, A126949, A274003.

is(n)=for(x=2,n-2,if(gcd(x,n)>1,next);my(t=Mod(x,n));while(abs(centerlift(t))>1,t*=x);if(t==-1,return(0)));n>1 \\ Charles R Greathouse IV, Aug 08 2013

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

A178751 Numbers k such that in Z/kZ the equation x^y + 1 = 0 has only the trivial solutions with x == -1 (mod k).

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