This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A086251 #37 Feb 16 2025 08:32:50
%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,
%T A086251 2,2,2,1,2,1,2,1,3,2,2,1,3,2,1,2,3,3,3,1,3,1,2,2,2,2,1,1,2,2,1,2,2,3,
%U A086251 1,2,3,2,3,2,2,3,1,1,3,1,3,2,2,2,1,1,2,2,1,1,3,4,1,2,3,2,2,1,3,3,2,3,2,2,3
%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.
%C A086251 Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - _Thomas Ordowski_, Sep 12 2017
%C A086251 The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - _Jeppe Stig Nielsen_, Sep 01 2020
%H A086251 <a href="/A086251/b086251.txt">Table of n, a(n) for n = 1..1206</a>
%H A086251 Eric Weisstein's World of Mathematics, <a href="https://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.
%F A086251 a(n) <= A182590(n). - _Thomas Ordowski_, Sep 14 2017
%F A086251 a(n) = A001221(A064078(n)). - _Thomas Ordowski_, Oct 26 2017
%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}]]
%o A086251 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ _Michel Marcus_, Sep 12 2017
%o A086251 (PARI) a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ _Jeppe Stig Nielsen_, Sep 01 2020
%Y A086251 Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.
%K A086251 hard,nonn
%O A086251 1,11
%A A086251 _T. D. Noe_, Jul 14 2003
%E A086251 Terms to a(500) in b-file from _T. D. Noe_, Nov 11 2010
%E A086251 Terms a(501)-a(1200) in b-file from _Charles R Greathouse IV_, Sep 14 2017
%E A086251 Terms a(1201)-a(1206) in b-file from _Max Alekseyev_, Sep 11 2022