A166226 Bell number n modulo n.

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, 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, 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

Offset: 1

Author

Thierry Banel (tbanel(AT)gmail.com), Oct 09 2009

Keywords

• nonn

Comments

a(n) = 2 (mod n) when n is prime.

Examples

			a(3)=a(5)=a(7)=a(11)=2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Greg Hurst, Andrew Schultz, An elementary (number theory) proof of Touchard's congruence, arXiv:0906.0696 [math.CO], (2009)

Crossrefs

See the Bell numbers sequence A000110.

Programs

Magma

[Bell(n) mod n: n in [1..100]]; // Vincenzo Librandi, Feb 03 2016

Maple

seq(combinat:-bell(n) mod n, n=1..100); # Robert Israel, Feb 03 2016

Mathematica

Array[n \[Function] Mod[BellB[n], n], 1000] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)
Table[Mod[BellB[n], n], {n, 1, 100}] (* G. C. Greubel, Feb 02 2016 *)

Formula

a(n) = A000110(n) mod n.

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

Extensions

More terms from R. J. Mathar and J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010