A086251 Number of primitive prime factors of 2^n - 1.

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, 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, 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

Offset: 1

Author

T. D. Noe, Jul 14 2003

Keywords

• hard
• nonn

Comments

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.

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

The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

Examples

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.
		

Links

• Table of n, a(n) for n = 1..1206
• Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Crossrefs

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.

Programs

Mathematica

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}]]

PARI

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

PARI

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

Formula

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.

a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017

a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Extensions

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010

Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017

Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022