A002982 - OEIS (original) (raw)

3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003

COMMENTS

The corresponding primes n!-1 are often called factorial primes.

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 719 at p. 160.

LINKS

Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Eric Weisstein's World of Mathematics, Factorial.

EXAMPLE

The sequence of numbers n! - 1 together with their prime indices begins:

1: {}

5: {3}

23: {9}

119: {4,7}

719: {128}

5039: {675}

40319: {9,273}

362879: {5,5,430}

3628799: {10,11746}

39916799: {6,7,9,992}

479001599: {25306287}

6227020799: {270,256263}

87178291199: {3610490805}

1307674367999: {7,11,11,16,114905}

20922789887999: {436,318519035}

355687428095999: {8,21,10165484947}

6402373705727999: {17,20157,25293727}

121645100408831999: {119,175195,4567455}

2432902008176639999: {11715,659539127675}

(End)

PROG

(Magma) [n: n in [0..500] | IsPrime(Factorial(n)-1)]; // Vincenzo Librandi, Sep 07 2017

(Python)

from sympy import factorial, isprime

A002982_list = [n for n in range(1, 10**2) if isprime(factorial(n)-1)] # Chai Wah Wu, Apr 04 2021

CROSSREFS

Cf. A002981 (numbers n such that n!+1 is prime).

Cf. A055490 (primes of form n!-1).

Cf. A088332 (primes of form n!+1).

EXTENSIONS

21480 sent in by Ken Davis (ken.davis(AT)softwareag.com), Oct 29 2001

Updated Feb 26 2007 by Max Alekseyev, based on progress reported in the Carmody web site.

Inserted missing 21480 and 34790 (see Caldwell). Added 94550, discovered Oct 05 2010. Eric W. Weisstein, Oct 06 2010

103040 was discovered by James Winskill, Dec 14 2010. It has 471794 digits. Corrected by Jens Kruse Andersen, Mar 22 2011

a(27) = 208003 from Sou Fukui, Jul 27 2016