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%I #27 Sep 08 2022 08:45:48
%S 0,0,2,3,2,5,2,4,6,5,2,1,2,12,5,3,2,13,2,12,15,5,2,9,3,18,10,3,2,27,2,
%T 12,4,5,0,1,2,24,28,27,2,23,2,8,5,5,2,33,24,20,49,39,2,5,27,28,34,5,2,
%U 57,2,36,6,51,47,19,2,52,15,25,2,49,2,42,22,71,59,19,2,44,23,5,2,65,84
%N Bell number n modulo n.
%C a(n) = 2 (mod n) when n is prime.
%H G. C. Greubel, <a href="/A166226/b166226.txt">Table of n, a(n) for n = 1..10000</a>
%H Greg Hurst, Andrew Schultz, <a href="http://arxiv.org/abs/0906.0696v2">An elementary (number theory) proof of Touchard's congruence</a>, arXiv:0906.0696 [math.CO], (2009)
%F a(n) = A000110(n) mod n.
%F a(p^m) = m+1 (mod p) when p is prime and m >= 1 (see Lemma 3.1 in the Hurst/Schultz reference). - _Joerg Arndt_, Jun 01 2016
%e a(3)=a(5)=a(7)=a(11)=2.
%p seq(combinat:-bell(n) mod n, n=1..100); # _Robert Israel_, Feb 03 2016
%t Array[n \[Function] Mod[BellB[n], n], 1000] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)
%t Table[Mod[BellB[n], n], {n, 1, 100}] (* _G. C. Greubel_, Feb 02 2016 *)
%o (Magma) [Bell(n) mod n: n in [1..100]]; _Vincenzo Librandi_, Feb 03 2016
%Y See the Bell numbers sequence A000110.
%K nonn
%O 1,3
%A Thierry Banel (tbanel(AT)gmail.com), Oct 09 2009
%E More terms from _R. J. Mathar_ and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010
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