A037153 a(n) = p-n!, where p is the smallest prime > n!+1.

2, 3, 5, 5, 7, 7, 11, 23, 17, 11, 17, 29, 67, 19, 43, 23, 31, 37, 89, 29, 31, 31, 97, 131, 41, 59, 47, 67, 223, 107, 127, 79, 37, 97, 61, 131, 311, 43, 97, 53, 61, 97, 71, 47, 239, 101, 233, 53, 83, 61, 271, 53, 71, 223, 71, 149, 107, 283, 293, 271, 769, 131, 271, 67, 193

Offset: 1

Author

Jud McCranie

Keywords

• nonn

Comments

Analogous to Fortunate numbers and like them, the entries appear to be primes. In fact, the first 1200 terms are primes. Are all terms prime?

a(n) is the first (smallest) m such that m > 1 and n!+ m is prime. The second such m is A087202(n). a(n) must be greater than nextprime(n)-1. - Farideh Firoozbakht, Sep 01 2003

Sequence A069941, which counts the primes between n! and n!+n^2, provides numerical evidence that the smallest prime p greater than n!+1 is a prime distance from n!; that is, p-n! is a prime number. For p-n! to be a composite number, p would have to be greater than n!+n^2, which would imply that A069941(n)=0. - T. D. Noe, Mar 06 2010

The first 4003 terms are prime. - Dana Jacobsen, May 10 2015

Links

• Ray Chandler and Dana Jacobsen, Table of n, a(n) for n = 1..4000 [first 1200 terms from Ray Chandler]
• Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
• Hisanori Mishima, Primes near to factorial, Dec 2008.
• Andy Nicol, Line graphs of A037153 in order and ascending numerical order

Crossrefs

Cf. A087202, A005235, A033932.

Programs

Magma

z:=125; [p-f where p is NextPrime(f+1) where f is Factorial(n): n in [1..z]]; // Klaus Brockhaus, Mar 02 2010

Mathematica

NextPrime[ n_Integer ] := (k=n+1; While[ !PrimeQ[ k ], k++ ]; Return[ k ]); f[ n_Integer ] := (p = n! + 1; q = NextPrime[ p ]; Return[ q - p + 1 ]); Table[ f[ n ], {n, 1, 75} ] (* Robert G. Wilson v *)

MuPAD

for n from 1 to 65 do f := n!:a := nextprime(f+2)-f:print(a) end_for; // Zerinvary Lajos, Feb 22 2007

PARI

a(n)=nextprime(n!+2)-n! \\ Charles R Greathouse IV, Jul 02 2013; Corrected by Dana Jacobsen, May 10 2015

Perl

use ntheory ":all"; for my $n (1..1000) { my $f=factorial($n); say "$n ",next_prime($f+1)-$f; } # Dana Jacobsen, May 10 2015

Python

from sympy import factorial, nextprime
def a(n): fn = factorial(n); return nextprime(fn+1) - fn
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, May 22 2022

Extensions

Edited by N. J. A. Sloane, Mar 06 2010