A097406 Largest primitive prime factor of 2^n-1, or a(n) = 1 if no such prime exists.

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 89, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 178481, 241, 1801, 2731, 262657, 113, 2089, 331, 2147483647, 65537, 599479, 43691, 122921, 109, 616318177, 174763, 121369, 61681, 164511353, 5419

Offset

1,2

Comments

By Zsigmondy's theorem, a(n) > 1 except for n = 1 or 6.

Conjectures: (1) For every n the highest unique prime factor is of the form kn+1. The values for k are in A097407. (2) For each composite n many factors of the form kn+1 occur intermittently but always singly in any cofactor pair. (3) For each prime n every factor is of the form kn+1.

A prime factor of 2^n-1 is called primitive if it does not divide 2^r-1 for any r<n, cf. A086251.

a(n) is the greatest prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists. - Jianing Song, Oct 23 2019

Links

Max Alekseyev, Table of n, a(n) for n = 1..1206

Formula

a(n) = A006530(A064078(n)). - Jianing Song, Oct 23 2019

Prog

(PARI) isprimitive(p, n) = {for (r=1, n-1, if (((2^r-1) % p) == 0, return (0)); ); return (1); }
a(n) = {f = factor(2^n-1); forstep(i=#f~, 1, -1, if (isprimitive(f[i, 1], n), return (f[i, 1])); ); return (1); } \\ Michel Marcus, Jul 15 2013

Crossrefs

Cf. A064078, A097407.

For the smallest primitive prime factor of 2^n-1 see A112927.

Sequence in context: A231609 A046561 A112927 * A064078 A292015 A342660

Adjacent sequences: A097403 A097404 A097405 * A097407 A097408 A097409

Keyword

nonn,easy

Author

Marco Matosic, Aug 16 2004

Extensions

More terms and better description from Vladeta Jovovic, Sep 03 2004

a(1) and a(6) changed from 0 to 1 by Jianing Song, Oct 23 2019

Status

approved