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A283300 Primes p such that p^2 divides Bell(p) - 2. 2
2, 5, 11, 109, 509, 4099, 2951209 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..7.

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

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A000110 (Bell numbers).

Sequence in context: A134998 A078790 A158999 * A069506 A239900 A123165

Adjacent sequences:  A283297 A283298 A283299 * A283301 A283302 A283303

KEYWORD

nonn,hard,more,nice

AUTHOR

Amiram Eldar, Mar 04 2017

EXTENSIONS

a(7) from Hiroaki Yamanouchi, Aug 30 2018

STATUS

approved

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Last modified July 4 15:43 EDT 2020. Contains 335448 sequences. (Running on oeis4.)