A168177 Number of prime factors of n! + 2^n - 1, counted with multiplicity.

1, 1, 1, 2, 1, 4, 1, 4, 2, 4, 1, 5, 4, 4, 4, 4, 2, 7, 2, 8, 4, 3, 3, 7, 3, 3, 5, 6, 5, 7, 3, 7, 5, 6, 3, 10, 5, 8, 4, 8, 7, 7, 9, 6, 5, 5, 4, 11, 5, 6, 3, 6, 4, 9, 6, 11, 5, 2, 4, 15, 2, 6, 5, 8, 5, 7, 4, 5, 7, 9, 2, 13, 7, 5, 8, 6, 4, 7, 3, 11, 5, 3, 3, 11, 6

Offset

1,4

Links

Kevin P. Thompson, Table of n, a(n) for n = 1..106

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) = A001222(A127986(n)). - Amiram Eldar, Feb 05 2020

Example

6! + 2^6 - 1 = 783 = 3^3 * 29, hence a(6) = 4.

Mathematica

PrimeOmega @ Table[n! + 2^n - 1, {n, 1, 30}] (* Amiram Eldar, Feb 05 2020 *)

Prog

(Magma) pfmult := func< n | &+[ d[2]: d in Factorization(n) ] >; [ pfmult(Factorial(n)+2^n-1): n in [1..50] ]; //Some values were computed using Dario Alpern's ECM Factorization Program.
(PARI) a(n)=bigomega(n!+2^n-1) \\ Charles R Greathouse IV, Feb 01 2013

Crossrefs

Cf. A001222, A127986 (n!+2^n-1), A139024 (number of distinct prime factors), A139023 (smallest prime factor), A127987 (largest prime factor).

Sequence in context: A193306 A053578 A368201 * A216864 A337175 A263432

Adjacent sequences: A168174 A168175 A168176 * A168178 A168179 A168180

Keyword

nonn

Author

Klaus Brockhaus, Nov 20 2009

Extensions

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

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

Status

approved