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A097406
Largest primitive prime factor of 2^n-1, or a(n) = 1 if no such prime exists.
6
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
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
For the smallest primitive prime factor of 2^n-1 see A112927.
Sequence in context: A231609 A046561 A112927 * A064078 A292015 A342660
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