%I A086251
%S A086251 0,1,1,1,1,0,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,2,3,1,1,1,1,1,2,
%T A086251 2,2,1,2,1,2,1,3,2,2,1,3,2,1,2,3,3,3,1,3,1,2,2,2,2,1,1,2,2,1,2,2,3,1,2,
%U A086251 3,2,3,2,2,3,1,1,3,1,3,2,2,2,1,1,2,2,1,1,3,4,1,2,3,2,2,1,3,3,2,3,2,2,3
%N A086251 Number of primitive prime factors of 2^n-1.
%C A086251 A prime factor of 2^n-1 is called primitive if it does not divide 2^r-1 
               for any r<n. Equivalently, p is a primitive prime factor of 2^n-1 
               if ord(2,p)=n. Zsigmondy's theorem says that there is at least one 
               primitive prime factor for n > 1, except for n=6. See A086252 for 
               those n that have a record number of primitive prime factors.
%H A086251 T. D. Noe, <a href="b086251.txt">Table of n, a(n) for n=1..500</a> (using 
               data from A001265)
%H A086251 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ZsigmondyTheorem.html">Zsigmondy Theorem</a>
%F A086251 a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
%e A086251 a(11) = 2 because 2^11-1 = 23*89 and both 23 and 89 have order 11.
%t A086251 Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, 
               f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]
%Y A086251 Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), 
               A086252.
%Y A086251 Sequence in context: A043286 A043285 A146291 this_sequence A092931 A147300 
               A110503
%Y A086251 Adjacent sequences: A086248 A086249 A086250 this_sequence A086252 A086253 
               A086254
%K A086251 hard,nonn
%O A086251 1,11
%A A086251 T. D. Noe (noe(AT)sspectra.com), Jul 14 2003