A214810 -id:A214810 - cp's OEIS frontend

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

Eric W. Weisstein, Feb 09 2002

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

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 et al., 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.

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

(* 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 *)

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

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

A214811 Triangle read by rows: row n lists prime factors of (p^p-1)/(p-1) where p = prime(n).

Original entry on oeis.org

3, 13, 11, 71, 29, 4733, 15797, 1806113, 53, 264031, 1803647, 10949, 1749233, 2699538733, 109912203092239643840221, 461, 1289, 831603031789, 1920647391913, 59, 16763, 84449, 2428577, 14111459, 58320973, 549334763, 568972471024107865287021434301977158534824481, 149, 1999, 7993, 16651, 17317, 10192715656759, 41903425553544839998158239

Offset: 1

N. J. A. Sloane, Jul 31 2012

			Triangle begins:
[3]
[13]
[11, 71]
[29, 4733]
[15797, 1806113]
[53, 264031, 1803647]
[10949, 1749233, 2699538733]
[109912203092239643840221]
[461, 1289, 831603031789, 1920647391913]
[59, 16763, 84449, 2428577, 14111459, 58320973, 549334763]
[568972471024107865287021434301977158534824481]
[149, 1999, 7993, 16651, 17317, 10192715656759, 41903425553544839998158239]
...

J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423. See Table 3.

Cf. A001039, A054767, A125135, A214810, A214812.

f:=proc(n) local i,t1,p,B,F;
p:=ithprime(n);
B:=(p^p-1)/(p-1);
F:=ifactors(B)[2];
lprint(n,p,B,F);
t1:=[seq(F[i][1],i=1..nops(F))];
sort(t1);
end;

A248843 Table read by rows in which row n lists divisors of (p^p-1)/(p-1) where p = prime(n).

Original entry on oeis.org

1, 3, 1, 13, 1, 11, 71, 781, 1, 29, 4733, 137257, 1, 15797, 1806113, 28531167061, 1, 53, 264031, 1803647, 13993643, 95593291, 476218721057, 25239592216021, 1, 10949, 1749233, 2699538733, 19152352117, 29557249587617, 4722122236541789

Offset: 1

Jean-François Alcover, Dec 03 2014

			Table begins:
  [1, 3],
  [1, 13],
  [1, 11, 71, 781],
  [1, 29, 4733, 137257],
  [1, 15797, 1806113, 28531167061],
  [1, 53, 264031, 1803647, 13993643, 95593291, 476218721057, 25239592216021],
  ...

Samuel S. Wagstaff, Aurifeuillian Factorizations and the Period of the Bell Numbers, Math. Comp. 65 (1996), 383-391.
Eric Weisstein's MathWorld, Bell Number
Wikipedia, Bell Number

Cf. A000110, A001039, A054767, A125135, A214810, A214811, A214812.

Table[p = Prime[n]; Divisors[(p^p - 1)/(p - 1)], {n, 1, 10}] // Flatten

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

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A214811 Triangle read by rows: row n lists prime factors of (p^p-1)/(p-1) where p = prime(n).

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A248843 Table read by rows in which row n lists divisors of (p^p-1)/(p-1) where p = prime(n).

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