cp's OEIS Frontend

A242292 Least prime divisor of 2^n - n which does not divide any 2^k - k with 0 < k < n, or 1 if such a primitive prime divisor of 2^n - n does not exist.

%I A242292 #11 Aug 05 2019 04:07:01
%S A242292 1,2,5,3,1,29,11,31,503,13,7,1021,8179,1637,4679,1,8737,131063,524269,
%T A242292 262139,2097131,349,131,773,271,197,457,1493,317,17,6733,73,41,157109,
%U A242292 83,53,1741,3329,49977801259,997,149,2199023255531,61,4398046511093,3769453
%N A242292 Least prime divisor of 2^n - n which does not divide any 2^k - k with 0 < k < n, or 1 if such a primitive prime divisor of 2^n - n does not exist.
%C A242292 Conjecture: a(n) = 1 only for n = 1, 5, 16.
%C A242292 In constrast, a classical theorem of Bang asserts that if n > 1 is different from 6 then 2^n - 1 has a prime divisor which does not divide any 2^k - 1 with 0 < k < n.
%D A242292 A. S. Bang, Taltheoretiske Undersgelser, Tidsskrift fur Mat. 4(1886), no. 5, 70--80, 130--137.
%H A242292 Zhi-Wei Sun, <a href="/A242292/b242292.txt">Table of n, a(n) for n = 1..215</a>
%H A242292 Z.-W. Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
%e A242292 a(4) = 3 since 2^4 - 4 = 2^2*3 with 3 dividing none of 2^1 - 1 = 1, 2^2 - 2 = 2 and 2^3 - 3 = 5.
%t A242292 u[n_]:=2^n-n
%t A242292 f[n_]:=FactorInteger[u[n]]
%t A242292 p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
%t A242292 Do[If[u[n]<2, Goto[cc]]; Do[Do[If[Mod[u[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 45}]
%Y A242292 Cf. A000040, A000325, A112927, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207.
%K A242292 nonn
%O A242292 1,2
%A A242292 _Zhi-Wei Sun_, May 10 2014