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 A054767 Period of the sequence of Bell numbers A000110 (mod n). 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. - T. D. Noe, Dec 17 2008 LINKS 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 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. - Franklin T. Adams-Watters, Jun 06 2006 MATHEMATICA (* 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], {k, 2, km}]]]; a = 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[] == 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 *) CROSSREFS Cf. A000110, A023037, A214810. Sequence in context: A107733 A273076 A272825 * A137947 A168437 A076747 Adjacent sequences:  A054764 A054765 A054766 * A054768 A054769 A054770 KEYWORD nonn,hard,nice AUTHOR Eric W. Weisstein, Feb 09 2002 EXTENSIONS 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 STATUS approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)