A146122 -id:A146122 - 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

A146093 Bell numbers (A000110) read mod 3.

Original entry on oeis.org

1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2, 1, 0, 0, 1, 0, 1, 1

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500
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.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110, A146094-A146122. A054767 gives periods. See also A002034.

[Bell(n) mod 3: n in [0..100]]; // Vincenzo Librandi, Jan 30 2016

Mod[BellB[Range[0,110]],3] (* Harvey P. Dale, Oct 31 2013 *)

a(n+13) = a(n). - G. C. Greubel, Jan 30 2016

A146094 Bell numbers (A000110) read mod 4.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0, 3, 3, 2, 1, 1, 2, 1, 3, 0, 3, 1, 0

Offset: 0

N. J. A. Sloane, Feb 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..2500
P. Pleasants, W. Lunnon, and N. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 4: n in [0..100]]; // Vincenzo Librandi, Jan 30 2016

Mod[BellB[Range[0, 50]], 4] (* G. C. Greubel, Jan 30 2016 *)

a(n) = a(n-12). - Charles R Greathouse IV, Jul 06 2011 [See A054767. - Jianing Song, Jun 20 2025]

A146121 Bell numbers (A000110) read mod 31.

Original entry on oeis.org

1, 1, 2, 5, 15, 21, 17, 9, 17, 5, 4, 11, 15, 1, 20, 30, 16, 8, 1, 21, 3, 25, 26, 19, 16, 2, 15, 16, 17, 12, 3, 2, 3, 7, 20, 5, 7, 26, 26, 22, 9, 15, 26, 16, 21, 19, 15, 24, 9, 22, 24, 28, 20, 14, 4, 18, 17, 0, 2, 29, 15, 5, 5, 10, 27, 25, 12, 2, 21, 17, 0, 24, 10, 11, 6, 9, 3, 8, 2, 0, 15, 21

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

Conjecture: a(n+568972471024107865287021434301977158534824481) = a(n). - G. C. Greubel, Feb 02 2016 [Conjecture is correct; see A054767. - Jianing Song, Jun 20 2025]

G. C. Greubel, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, order 568972471024107865287021434301977158534824481.

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 31: n in [0..100]]; // G. C. Greubel, Feb 02 2016

Mod[BellB[Range[0, 100]], 31] (* G. C. Greubel, Feb 02 2016 *)

A146095 Bell numbers (A000110) read mod 5.

Original entry on oeis.org

1, 1, 2, 0, 0, 2, 3, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 4, 4, 2, 2, 1, 3, 1, 4, 3, 4, 4, 0, 2, 2, 3, 4, 2, 4, 0, 2, 1, 1, 4, 2, 3, 2, 0, 1, 0, 0, 2, 1, 1, 0, 2, 3, 2, 1, 2, 0, 0, 3, 3, 2, 0, 3, 1, 0, 2, 3, 4, 1, 2, 0, 2, 0, 3, 2, 2, 2, 3, 0, 4, 4, 0, 3, 4, 3, 4, 3, 2, 2, 2, 2, 0, 4, 4, 4, 2, 4, 3, 3, 1, 1, 2, 1, 4, 2

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500
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.
Index entries for linear recurrences with constant coefficients, order 781.

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 5: n in [0..100]]; // Vincenzo Librandi, Jan 30 2016

Mod[BellB[Range[0, 50]], 5] (* G. C. Greubel, Jan 30 2016 *)

a(n) = a(n-781). - Charles R Greathouse IV, Jul 06 2011 [See A054767. - Jianing Song, Jun 20 2025]

A146096 Bell numbers (A000110) read mod 6.

Original entry on oeis.org

1, 1, 2, 5, 3, 4, 5, 1, 0, 3, 1, 0, 1, 1, 4, 5, 5, 0, 1, 5, 4, 3, 3, 4, 3, 1, 4, 1, 5, 2, 3, 1, 2, 1, 3, 0, 1, 3, 4, 1, 1, 2, 5, 3, 4, 5, 1, 0, 3, 1, 0, 1, 1, 4, 5, 5, 0, 1, 5, 4, 3, 3, 4, 3, 1, 4, 1, 5, 2, 3, 1, 2, 1, 3, 0, 1, 3, 4, 1, 1, 2, 5, 3, 4, 5, 1, 0, 3, 1, 0, 1, 1, 4, 5, 5, 0, 1, 5, 4, 3, 3, 4, 3, 1, 4

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500
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.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 6: n in [0..100]]; // Vincenzo Librandi, Jan 30 2016

Mod[BellB[Range[0, 50]], 6] (* G. C. Greubel, Jan 30 2016 *)

a(n+39) = a(n), for n>=0. - G. C. Greubel, Jan 30 2016 [See A054767. - Jianing Song, Jun 20 2025]

A146097 Bell numbers (A000110) read mod 7.

Original entry on oeis.org

1, 1, 2, 5, 1, 3, 0, 2, 3, 0, 6, 4, 3, 2, 5, 3, 6, 3, 0, 5, 0, 1, 2, 2, 3, 5, 5, 1, 3, 4, 5, 1, 3, 6, 4, 0, 2, 6, 4, 2, 3, 4, 2, 1, 3, 6, 5, 0, 6, 3, 4, 2, 4, 5, 6, 2, 0, 6, 6, 2, 4, 1, 2, 6, 5, 1, 6, 5, 3, 1, 4, 6, 0, 4, 1, 4, 5, 3, 6, 4, 5, 5, 2, 1, 2, 3, 2, 3, 0, 3, 3, 5, 5, 5, 3, 3, 6, 1, 3, 3, 1, 6, 2, 0, 4

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, order 137257.

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 7: n in [0..100]]; // Vincenzo Librandi, Feb 01 2016

seq(combinat:-bell(n) mod 7, n=0..100); # Robert Israel, Jan 31 2016

Mod[BellB[Range[0, 100]], 7] (* G. C. Greubel, Jan 31 2016 *)

a(n+137257) = a(n). - G. C. Greubel, Jan 31 2016 [See A054767. - Jianing Song, Jun 20 2025]

A146098 Bell numbers (A000110) read mod 8.

Original entry on oeis.org

1, 1, 2, 5, 7, 4, 3, 5, 4, 3, 7, 2, 5, 5, 2, 1, 3, 4, 7, 1, 4, 7, 3, 2, 1, 1, 2, 5, 7, 4, 3, 5, 4, 3, 7, 2, 5, 5, 2, 1, 3, 4, 7, 1, 4, 7, 3, 2, 1, 1, 2, 5, 7, 4, 3, 5, 4, 3, 7, 2, 5, 5, 2, 1, 3, 4, 7, 1, 4, 7, 3, 2, 1, 1, 2, 5, 7, 4, 3, 5, 4, 3, 7, 2, 5, 5, 2, 1, 3, 4, 7, 1, 4, 7, 3, 2, 1, 1, 2, 5, 7, 4, 3, 5, 4

Offset: 0

N. J. A. Sloane, Feb 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 8: n in [0..100]]; // Vincenzo Librandi, Feb 01 2016

Mod[BellB[Range[0, 100]], 8] (* G. C. Greubel, Jan 31 2016 *)

a(n) = a(n-24). [Charles R Greathouse IV, Jul 06 2011] [See A054767. - Jianing Song, Jun 20 2025]

A146099 Bell numbers (A000110) read mod 9.

Original entry on oeis.org

1, 1, 2, 5, 6, 7, 5, 4, 0, 6, 1, 6, 4, 1, 7, 5, 5, 0, 4, 5, 1, 6, 6, 1, 6, 4, 1, 1, 5, 2, 6, 4, 5, 4, 6, 3, 1, 6, 1, 1, 1, 2, 5, 6, 7, 5, 4, 0, 6, 1, 6, 4, 1, 7, 5, 5, 0, 4, 5, 1, 6, 6, 1, 6, 4, 1, 1, 5, 2, 6, 4, 5, 4, 6, 3, 1, 6, 1, 1, 1, 2, 5, 6, 7, 5, 4, 0, 6, 1, 6, 4, 1, 7, 5, 5, 0, 4, 5, 1, 6, 6, 1, 6, 4, 1

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 9: n in [0..100]]; // Vincenzo Librandi, Feb 01 2016

Mod[BellB[Range[0, 100]], 9] (* G. C. Greubel, Jan 31 2016 *)

a(n+39) = a(n). - G. C. Greubel, Jan 31 2016 [See A054767. - Jianing Song, Jun 20 2025]

A146100 Bell numbers (A000110) read mod 10.

Original entry on oeis.org

1, 1, 2, 5, 5, 2, 3, 7, 0, 7, 5, 0, 7, 7, 2, 5, 7, 4, 9, 7, 2, 1, 3, 6, 9, 3, 4, 9, 5, 2, 7, 3, 4, 7, 9, 0, 7, 1, 6, 9, 7, 8, 7, 5, 6, 5, 5, 2, 1, 1, 0, 7, 3, 2, 1, 7, 0, 5, 3, 8, 7, 5, 8, 1, 5, 2, 3, 9, 6, 7, 5, 2, 5, 3, 2, 7, 7, 8, 5, 9, 4, 5, 3, 4, 3, 9, 8, 7, 7, 2, 7, 5, 4, 9, 9, 2, 9, 3, 8, 1, 1, 2, 1, 9, 2

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

Sequence is of period A054767(10) = 2343. - Jianing Song, Jun 20 2025

G. C. Greubel, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, order 2343.

Cf. A000110, A146093-A146122 (Bell numbers read mod 3 to mod 32).

Cf. A054767 (periods).

[Bell(n) mod 10: n in [0..100]]; // Vincenzo Librandi, Feb 01 2016

Mod[BellB[Range[0, 100]], 10] (* G. C. Greubel, Jan 31 2016 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A146093 Bell numbers (A000110) read mod 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A146094 Bell numbers (A000110) read mod 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A146121 Bell numbers (A000110) read mod 31.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A146095 Bell numbers (A000110) read mod 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A146096 Bell numbers (A000110) read mod 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A146097 Bell numbers (A000110) read mod 7.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A146098 Bell numbers (A000110) read mod 8.

Views

Author

Keywords

Links

Crossrefs

Programs