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A283300
Primes p such that p^2 divides Bell(p) - 2.
2
2, 5, 11, 109, 509, 4099, 2951209
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
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
KEYWORD
nonn,hard,more,nice
AUTHOR
Amiram Eldar, Mar 04 2017
EXTENSIONS
a(7) from Hiroaki Yamanouchi, Aug 30 2018
STATUS
approved