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A054767
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Period of the sequence of Bell numbers A000110 (mod n).
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4
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1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 117, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 117
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| For p prime, a(p) divides (p^p-1)/(p-1) = A023037(p), with equality at least for p up to 19.
Wagstaff shows that N(p) = (p^p-1)/(p-1) is the period for all primes p < 102 and for primes p = 113, 163, 167 and 173. For p = 7547, N(p) is a probable prime, which means that this p may have the maximum possible period N(p) also. See A088790. [From T. D. Noe (noe(AT)sspectra.com), Dec 17 2008]
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REFERENCES
| J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. [From N. J. A. Sloane (njas(AT)research.att.com), Feb 07 2009]
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Samuel S. Wagstaff Jr., Aurifeuillian factorizations and the period of the Bell numbers modulo a prime, Math. Comp. 65 (1996), 383-391. [From T. D. Noe (noe(AT)sspectra.com), Dec 17 2008]
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FORMULA
| If gcd(n,m) = 1, a(n*m) = lcm(a(n), a(m)). But the sequence is not in general multiplicative; e.g. a(2) = 3, a(9) = 39 and a(18) = 39. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 06 2006
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CROSSREFS
| Cf. A000110, A023037.
Sequence in context: A142351 A121565 A107733 * A137947 A168437 A076747
Adjacent sequences: A054764 A054765 A054766 * A054768 A054769 A054770
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KEYWORD
| nonn,hard,nice
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Feb 09, 2002
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EXTENSIONS
| More information from Phil Carmody (pc+oeis(AT)asdf.org), Dec 22 2002
Extended by T. D. Noe (noe(AT)sspectra.com), Dec 18 2008
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