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A054767 Period of the sequence of Bell numbers A000110 (mod n).

%I A054767 #47 Jun 19 2025 20:57:54
%S A054767 1,3,13,12,781,39,137257,24,39,2343,28531167061,156,25239592216021,
%T A054767 411771,10153,48,51702516367896047761,39,109912203092239643840221,
%U A054767 9372,1784341,85593501183,949112181811268728834319677753,312,3905,75718776648063,117,1647084
%N A054767 Period of the sequence of Bell numbers A000110 (mod n).
%C A054767 For p prime, a(p) divides (p^p-1)/(p-1) = A023037(p), with equality at least for p up to 19.
%C A054767 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
%H A054767 Jianing Song, <a href="/A054767/b054767.txt">Table of n, a(n) for n = 1..102</a> (b-file based on the Wagstaff article)
%H A054767 J. Levine and R. E. Dalton, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0148604-2">Minimum Periods, Modulo p, of First Order Bell Exponential Integrals</a>, Mathematics of Computation, 16 (1962), 416-423.
%H A054767 W. F. Lunnon et al., <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3511.pdf">Arithmetic properties of Bell numbers to a composite modulus I</a>, Acta Arithmetica 35 (1979), pp. 1-16.
%H A054767 Jianing Song, <a href="/A054767/a054767.pdf">A summary of several proofs in the Lunnon article</a>.
%H A054767 Samuel S. Wagstaff Jr., <a href="https://doi.org/10.1090/S0025-5718-96-00683-7">Aurifeuillian factorizations and the period of the Bell numbers modulo a prime</a>, Math. Comp. 65 (1996), 383-391.
%H A054767 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellNumber.html">Bell Number</a>.
%F A054767 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
%F A054767 a(2^s) = 3*2^s for s >= 2 (Theorem 6.4 in the Lunnon article). For an odd prime p, if a(p) = (p^p-1)/(p-1) (which is conjectured to hold for all p), then a(p^s) = p^(s-1)*(p^p-1)/(p-1) (Theorem 6.2 in the Lunnon article). - _Jianing Song_, Jun 18 2025
%t A054767 (* Warning: this program is just a verification of the existing data
%t A054767  and should not be used to extend the sequence beyond a(28) *)
%t A054767 BellMod[k_, m_] := Mod[Sum[Mod[StirlingS2[k, j], m], {j, 1, k}], m];
%t A054767 BellMod[k_, 1] := BellB[k];
%t A054767 period[nn_List] := Module[{lgmin=2, lgmax=5, nn1},
%t A054767    lg=If[Length[nn]<=lgmax, lgmin, lgmax];
%t A054767    nn1 = nn[[1;;lg]];
%t A054767    km=Length[nn]-lg;
%t A054767    Catch[Do[If[nn1==nn[[k;;k+lg-1]], Throw[k-1]];
%t A054767    If[k==km, Throw[0]], {k, 2, km}]]];
%t A054767 dd[n_] := SelectFirst[Table[{d, n/d},
%t A054767      {d, Divisors[n][[2;;-2]]}], GCD@@#==1&];
%t A054767 a[1]=1;
%t A054767 a[p_?PrimeQ] := a[p] = (p^p-1)/(p-1);
%t A054767 a[n_/;n>4 && dd[n]!={}] := With[{g = dd[n]}, LCM[a[g[[1]]], a[g[[2]]]]];
%t A054767 a[n_/;MemberQ[FactorInteger[n][[All, 2]], 1]] := a[n]=
%t A054767    With[{pp = Select[FactorInteger[n], #1[[2]] ==1 &][[All, 1]]},
%t A054767       a[n/Times@@pp]*Times@@a/@pp];
%t A054767 a[n_/;n>4 && GCD @@ FactorInteger[n][[All, 2]]>1] := a[n]=
%t A054767    With[{g=GCD @@ FactorInteger[n][[All, 2]]}, n^(1/g)*a[n^(1-1/g)]];
%t A054767 a[n_] := period[Table[BellMod[k, n], {k, 1, 28}]];
%t A054767 Table[a[n], {n, 1, 28}] (* _Jean-François Alcover_, Jul 31 2012, updated May 06 2024 *)
%Y A054767 Cf. A000110, A023037, A214810. A146093-A146122 gives Bell numbers read mod 3 to mod 32.
%K A054767 nonn,hard,nice
%O A054767 1,2
%A A054767 _Eric W. Weisstein_, Feb 09 2002
%E A054767 More information from _Phil Carmody_, Dec 22 2002
%E A054767 Extended by _T. D. Noe_, Dec 18 2008
%E A054767 a(26) corrected by _Jean-François Alcover_, Jul 31 2012
%E A054767 a(18) corrected by _Charles R Greathouse IV_, Jul 31 2012
%E A054767 a(27)-a(28) from _Charles R Greathouse IV_, Sep 07 2016