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A286635 a(n) is the smallest composite (pseudoprime) p such that Bell(n+p) == Bell(n)+Bell(n+1) (mod p). 0
21361, 8, 4, 134, 6, 4, 57, 6, 34, 65, 14, 9, 14, 6, 4, 21, 12, 4, 26, 8, 26, 779, 102, 99, 33, 8, 4, 14, 12, 4, 9, 6, 70, 33, 169, 25, 98, 8, 4, 14, 410, 4, 458, 6, 10, 25, 6, 26, 14, 8, 4, 122, 6, 4, 231, 8, 836, 62, 18, 74, 39, 8, 4, 1101, 14, 4, 81, 8, 68, 9, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Jacques Touchard proved in 1933 that for the Bell numbers (A000110), Bell(p+k) == Bell(k+1) + Bell(k) (mod p) for all primes p and k >= 0.
a(0)=21361 is the smallest pseudoprime of the congruence Bell(p) == 2(mod p). It was found by W. F. Lunnon and verified to be the smallest by David W. Wilson in 2007 (see comment in A000110).
a(84) is the first term that is larger than a(0).
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
a(1)=8 since 8 is composite, yet Bell(8+1)-Bell(1)-Bell(2) = 21144 = 8 * 3 * 881 is divisible by 8
CROSSREFS
Cf. A000110.
Sequence in context: A250532 A234454 A319017 * A324259 A083361 A236661
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 12 2017
STATUS
approved

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Last modified February 29 19:20 EST 2024. Contains 370428 sequences. (Running on oeis4.)