A068499 - OEIS

%I #54 Aug 27 2024 18:15:11

%S 1,2,3,4,6,10,12,16,18,22,28,30,36,40,42,46,52,58,60,66,70,72,78,82,

%T 88,96,100,102,106,108,112,126,130,136,138,148,150,156,162,166,172,

%U 178,180,190,192,196,198,210,222,226,228,232,238,240,250,256,262,268,270

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

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

%C 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]

%C 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

%C 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

%C a(n) = A039915(n-1) for 3 < n <= 1000. - _Georg Fischer_, Oct 19 2018

%H Jean-Christophe Hervé, <a href="/A068499/b068499.txt">Table of n, a(n) for n = 1..10000</a>

%H David J. Hemmer and Karlee J. Westrem, <a href="https://arxiv.org/abs/2402.02250">Palindrome Partitions and the Calkin-Wilf Tree</a>, arXiv:2402.02250 [math.CO], 2024. See Remark 3.3 p. 6.

%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>

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

%F For n <> 3, all terms are one less prime. - _Zak Seidov_, Aug 22 2012

%F a(n) = Integer part of A078456(n+1)/A078456(n). - _Eric Desbiaux_, May 07 2013

%e Illustration of the sieve: keep 1 = a(1) and then

%e 1st step: take 2 = a(2) and cross off 5, 8, 11, 14, 17, 20, 23, 26, etc.

%e 2nd step: take 3 = a(3) and cross off 7, 11, 15, 19, 23, 27, etc.

%e 3rd step: take 4 = a(4) and cross off 9, 14, 19, 24, etc.

%e 4th step: take 6 = a(5) and cross off 13, 19, 25 etc.

%e 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

%t Select[Range[300],Mod[#!,#+1]!=0&] (* _Harvey P. Dale_, Apr 11 2012 *)

%o (PARI) {plnt=1 ; nfa=1; mxind=60 ;  for(k=1, 10^7, nfa=nfa*k;

%o if(nfa % (k+1) != 0 , print1(k, ", "); plnt++ ;

%o if(mxind <  plnt, break() )))} \\ _Douglas Latimer_, Apr 25 2012

%o (PARI) a(n)=if(n<5,n,prime(n-1)-1) \\ _Charles R Greathouse IV_, Apr 25 2012

%o (Python)

%o from sympy import prime

%o def A068499(n): return prime(n-1)-1 if n>3 else n # _Chai Wah Wu_, Aug 27 2024

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

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Mar 11 2002