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

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

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

Jianing Song, Table of n, a(n) for n = 1..102 (b-file based on the Wagstaff article)

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, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.

Jianing Song, A summary of several proofs in the Lunnon article.

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

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

In particular, a(p^s) divides p^(s-1)*(p^p-1)/(p-1) for all odd prime p, and the exponent of p on the right is optimal. See Theorem 6.2 in the Lunnon, Pleasants, & Stephens link, and note that it uses umbral notation. - Charles R Greathouse IV, Feb 02 2026

Mathematica

(* Warning: this program is just a verification of the existing data
 and should not be used to extend the sequence beyond a(28) *)
BellMod[k_, m_] := Mod[Sum[Mod[StirlingS2[k, j], m], {j, 1, k}], m];
BellMod[k_, 1] := BellB[k];
period[nn_List] := 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}]]];
dd[n_] := SelectFirst[Table[{d, n/d},
     {d, Divisors[n][[2;; -2]]}], GCD@@#==1&];
a[1]=1;
a[p_?PrimeQ] := a[p] = (p^p-1)/(p-1);
a[n_/; n>4 && dd[n]!={}] := With[{g = dd[n]}, LCM[a[g[[1]]], a[g[[2]]]]];
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, 28}]];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jul 31 2012, updated May 06 2024 *)

Crossrefs

Cf. A000110, A023037, A214810. A146093-A146122 gives Bell numbers read mod 3 to mod 32.

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