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A054767
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Period of the sequence of Bell numbers A000110 (mod n).
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10
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1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 39, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 75718776648063, 117, 1647084
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OFFSET
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1,2
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COMMENTS
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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. - T. D. Noe, Dec 17 2008
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LINKS
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Table of n, a(n) for n=1..28.
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 Arithmetica 35 (1979), pp. 1-16.
Samuel S. Wagstaff Jr., Aurifeuillian factorizations and the period of the Bell numbers modulo a prime, Math. Comp. 65 (1996), 383-391.
Eric Weisstein's World of Mathematics, Bell Number
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FORMULA
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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. - Franklin T. Adams-Watters, Jun 06 2006
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MATHEMATICA
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(* n.b. this program might be incorrect beyond a(26) *) BellMod[k_, m_] := Mod[ Sum[ Mod[ StirlingS2[k, j], m], {j, 1, k}], m]; BellMod[k_, 1] := BellB[k]; period[nn_] := Module[{lgmin = 2, lgmax = 5, nn1}, lg = If[Length[nn] <= lgmax, lgmin, lgmax]; nn1 = nn[[1 ;; lg]]; km = Length[nn] - lg; Catch[ Do[ If[ nn1 == nn[[k ;; k + lg - 1]], Throw[k - 1]]; If[k == km, Throw[0]], {k, 2, km}]]]; a[1] = 1; a[p_?PrimeQ] := (p^p - 1)/(p - 1); a[n_ /; MemberQ[ FactorInteger[n][[All, 2]], 1]] := a[n] = With[{pp = Select[ FactorInteger[n], #1[[2]] == 1 & ][[All, 1]]}, a[n/Times @@ pp]*Times @@ a /@ pp]; a[n_ /; n > 4 && GCD @@ FactorInteger[n][[All, 2]] > 1] := a[n] = With[{g = GCD @@ FactorInteger[n][[All, 2]]}, n^(1/g)*a[n^(1 - 1/g)]]; a[n_] := period[ Table[ BellMod[k, n], {k, 1, 18}]]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Jul 31 2012 *)
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CROSSREFS
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Cf. A000110, A023037, A214810.
Sequence in context: A107733 A273076 A272825 * A137947 A168437 A076747
Adjacent sequences: A054764 A054765 A054766 * A054768 A054769 A054770
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KEYWORD
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nonn,hard,nice
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AUTHOR
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Eric W. Weisstein, Feb 09 2002
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EXTENSIONS
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More information from Phil Carmody, Dec 22 2002
Extended by T. D. Noe, Dec 18 2008
a(26) corrected by Jean-François Alcover, Jul 31 2012
a(18) corrected by Charles R Greathouse IV, Jul 31 2012
a(27)-a(28) from Charles R Greathouse IV, Sep 07 2016
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STATUS
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approved
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