cp's OEIS Frontend

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

%I A097406 #23 Apr 26 2022 22:17:13
%S A097406 1,3,7,5,31,1,127,17,73,11,89,13,8191,43,151,257,131071,19,524287,41,
%T A097406 337,683,178481,241,1801,2731,262657,113,2089,331,2147483647,65537,
%U A097406 599479,43691,122921,109,616318177,174763,121369,61681,164511353,5419
%N A097406 Largest primitive prime factor of 2^n-1, or a(n) = 1 if no such prime exists.
%C A097406 By Zsigmondy's theorem, a(n) > 1 except for n = 1 or 6.
%C A097406 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.
%C A097406 A prime factor of 2^n-1 is called primitive if it does not divide 2^r-1 for any r<n, cf. A086251.
%C A097406 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
%H A097406 Max Alekseyev, <a href="/A097406/b097406.txt">Table of n, a(n) for n = 1..1206</a>
%F A097406 a(n) = A006530(A064078(n)). - _Jianing Song_, Oct 23 2019
%o A097406 (PARI) isprimitive(p, n) = {for (r=1, n-1, if (((2^r-1) % p) == 0, return (0));); return (1);}
%o A097406 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
%Y A097406 Cf. A064078, A097407.
%Y A097406 For the smallest primitive prime factor of 2^n-1 see A112927.
%K A097406 nonn,easy
%O A097406 1,2
%A A097406 _Marco Matosic_, Aug 16 2004
%E A097406 More terms and better description from _Vladeta Jovovic_, Sep 03 2004
%E A097406 a(1) and a(6) changed from 0 to 1 by _Jianing Song_, Oct 23 2019