A283300 Primes p such that p^2 divides Bell(p) - 2.

2, 5, 11, 109, 509, 4099, 2951209

Offset: 1

Author

Amiram Eldar, Mar 04 2017

Keywords

• nonn
• hard
• more
• nice

Comments

A special case of Touchard's congruence is Bell(p) == 2 (mod p) for all primes p, where Bell(n) are the Bell numbers (A000110). These primes are for Touchard's congruence as Wieferich primes (A001220) are for Fermat's little theorem and Wilson primes (A007540) are for Wilson's theorem.

Examples

			For n=3, a(3)=11, Bell(11)=678570, Bell(11) - 2 = 11^2 * 61688.
		

References

• J. Touchard, "Propriétés arithmétiques de certains nombres récurrents", Ann. Soc. Sci. Bruxelles A 53 (1933), pp. 21-31.

Links

• Eric Weisstein's World of Mathematics, Touchard's Congruence

Crossrefs

Cf. A000110 (Bell numbers).

Programs

Mathematica

Select[Prime[Range[1000]], Divisible[BellB[#]-2, #^2] &]

Extensions

a(7) from Hiroaki Yamanouchi, Aug 30 2018