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A086257 Number of primitive prime factors of 2^n+1.

%I A086257 #44 Mar 15 2025 14:05:20
%S A086257 1,1,1,0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,1,2,2,3,1,1,2,2,1,2,2,
%T A086257 3,2,2,2,3,1,1,2,2,1,2,1,4,2,2,1,3,3,2,2,2,2,2,2,2,3,1,1,4,1,2,3,2,2,
%U A086257 2,2,2,2,2,2,4,1,3,3,4,1,2,3,4,5,2,1,4,1,3,3,3,3,1,2,3,2,1,4,3,2,4,1,4,2,1
%N A086257 Number of primitive prime factors of 2^n+1.
%C A086257 A prime factor of 2^n+1 is called primitive if it does not divide 2^r+1 for any r<n. Zsigmondy's theorem says that there is at least one primitive prime factor except for n=3. See A086258 for those n that have a record number of primitive prime factors.
%H A086257 Max Alekseyev, <a href="/A086257/b086257.txt">Table of n, a(n) for n = 0..1128</a> (terms 0..500 from T. D. Noe; terms 501..1062 from Amiram Eldar)
%H A086257 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ZsigmondyTheorem.html">Zsigmondy Theorem</a>
%F A086257 For n > 0, a(n) = A086251(2*n). - _Max Alekseyev_, Sep 06 2022
%e A086257 a(14) = 2 because 2^14+1 = 5*29*113 and 29 and 113 do not divide 2^r+1 for r < 14.
%t A086257 nMax=200; pLst={}; Table[f=Transpose[FactorInteger[2^n+1]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 0, nMax}]
%Y A086257 Excluding a(0) = 1, forms a bisection of A086251.
%Y A086257 Cf. A046799 (number of distinct prime factors of 2^n+1), A054992 (number of prime factors, with repetition, of 2^n+1), A086258.
%K A086257 hard,nonn
%O A086257 0,15
%A A086257 _T. D. Noe_, Jul 14 2003