A139024 Number of distinct prime factors of n! + 2^n - 1.

1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 1, 4, 4, 4, 4, 4, 2, 5, 2, 7, 3, 3, 3, 6, 3, 3, 5, 6, 5, 6, 3, 7, 5, 6, 3, 8, 5, 8, 4, 7, 7, 5, 9, 6, 5, 5, 4, 10, 5, 6, 3, 6, 4, 6, 6, 11, 5, 2, 4, 13, 2, 6, 4, 8, 5, 6, 4, 5, 7, 9, 2, 11, 7, 5, 8, 6, 4, 6, 3, 10, 5, 3, 3, 9, 6, 4

Offset

1,4

Links

Kevin P. Thompson, Table of n, a(n) for n = 1..103

Florian Luca and Igor E. Shparlinski, On the largest prime factor of n! + 2^n - 1, Journal de Théorie des Nombres de Bordeaux 17 (2005), 859-870.

Formula

a(n) = A001221(A127986(n)). - Amiram Eldar, Feb 05 2020

Example

a(6) = 2 since n! + 2^n - 1 = 6! + 2^6 - 1 = 783 = 3^3 * 29 has 2 distinct prime factors.

Mathematica

a = {}; Do[AppendTo[a, n! + 2^n - 1], {n, 1, 40}]; b = {}; Do[c = Length[FactorInteger[a[[n]]]]; AppendTo[b, c], {n, 1, Length[a]}]; b
PrimeNu @ Table[n! + 2^n - 1, {n, 1, 30}] (* Amiram Eldar, Feb 05 2020 *)

Prog

(PARI) a(n)=omega(n!+2^n-1) \\ Charles R Greathouse IV, Feb 01 2013

Crossrefs

Cf. A001221, A127986, A127987, A139023.

Cf. A168177. - From Klaus Brockhaus, Nov 19 2009

Sequence in context: A054065 A194868 A304574 * A217317 A331128 A154958

Adjacent sequences: A139021 A139022 A139023 * A139025 A139026 A139027

Keyword

nonn

Author

Artur Jasinski, Apr 06 2008, corrected Apr 22 2008

Extensions

a(1) - a(40) verified and a(41)- a(75) added by Klaus Brockhaus, Nov 19 2009

a(76)-a(81) from Amiram Eldar, Feb 05 2020

a(82)-a(86) from Kevin P. Thompson, Jun 29 2022

Status

approved