A068499 Numbers m such that m! reduced modulo (m+1) is not zero.

1, 2, 3, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

Author

Benoit Cloitre, Mar 11 2002

Keywords

• easy
• nonn

Comments

Also n such that tau((n+1)!) = 2* tau(n!)

For n > 2, a(n) is the smallest number such that a(n) !== -1 (mod a(k)+1) for any 1 < k < n. [Franklin T. Adams-Watters, Aug 07 2009]

Also n such that sigma((n+1)!) = (n+2)* sigma(n!), which is the same as A062569(n+1) = (n+2)*A062569(n). - Zak Seidov, Aug 22 2012

This sequence is obtained by the following sieve: keep 1 in the sequence and then, at the k-th step, keep the smallest number, x say, that has not been crossed off before and cross off all the numbers of the form k*(x + 1) - 1 with k > 1. The numbers that are left form the sequence. - Jean-Christophe Hervé, Dec 12 2015

a(n) = A039915(n-1) for 3 < n <= 1000. - Georg Fischer, Oct 19 2018

Examples

			Illustration of the sieve: keep 1 = a(1) and then
1st step: take 2 = a(2) and cross off 5, 8, 11, 14, 17, 20, 23, 26, etc.
2nd step: take 3 = a(3) and cross off 7, 11, 15, 19, 23, 27, etc.
3rd step: take 4 = a(4) and cross off 9, 14, 19, 24, etc.
4th step: take 6 = a(5) and cross off 13, 19, 25 etc.
10 is obtained at next step and the smallest crossed off numbers are then 21 and 28. This gives the beginning of the sequence up to 22 = a(10): 1, 2, 3, 4, 6, 10, 12, 16, 18, 22. - _Jean-Christophe Hervé_, Dec 12 2015
		

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000
• David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Remark 3.3, p. 6.
• Karlee J. Westrem, Schaper numbers, palindrome partitions, and symmetric functions, with applications to characters of the symmetric group, Ph. D. Dissertation, Michigan Tech. Univ. (2025). See p. 49.
• Index entries for sequences generated by sieves

Crossrefs

Cf. A000040, A039915, A062569, A166460 (almost complement).

Programs

Mathematica

Select[Range[300],Mod[#!,#+1]!=0&] (* Harvey P. Dale, Apr 11 2012 *)

PARI

{plnt=1 ; nfa=1; mxind=60 ;  for(k=1, 10^7, nfa=nfa*k;
if(nfa % (k+1) != 0 , print1(k, ", "); plnt++ ;
if(mxind <  plnt, break() )))} \\ Douglas Latimer, Apr 25 2012

PARI

a(n)=if(n<5,n,prime(n-1)-1) \\ Charles R Greathouse IV, Apr 25 2012

Python

from sympy import prime
def A068499(n): return prime(n-1)-1 if n>3 else n # Chai Wah Wu, Aug 27 2024

Formula

For n >= 4, a(n) = prime(n-1) - 1 = A006093(n-1).

For n <> 3, all terms are one less prime. - Zak Seidov, Aug 22 2012

a(n) = Integer part of A078456(n+1)/A078456(n). - Eric Desbiaux, May 07 2013