A054767 -id:A054767 - cp's OEIS frontend

A155730 Indices of Bell numbers that are divisible by 5.

3, 4, 8, 10, 11, 15, 28, 35, 43, 45, 46, 50, 56, 57, 61, 64, 70, 72, 78, 81, 91, 107, 109, 119, 126, 128, 135, 141, 147, 149, 158, 170, 179, 181, 187, 193, 208, 210, 220, 221, 223, 225, 236, 245, 254, 257, 263, 264, 268, 275, 276, 280, 286, 288, 297, 298, 300

Offset: 1

David Pasino, Jan 25 2009

nonn

First differences of terms of this sequence has a period of 156: a(156*m + n) - a(156*m + n -1) = a(156*s + n)- a(156*s + n - 1) for m and s >= 0. - Enrique Pérez Herrero, Oct 01 2013

Vaclav Kotesovec, Table of n, a(n) for n = 1..9990 (terms 1..1000 from Enrique Pérez Herrero).
J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423.
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, A229004, A054767, A016789.

Select[Range[300], Mod[BellB[#], 5] == 0 &] (* T. D. Noe, Oct 01 2013 *)

A229004 Indices of Bell numbers divisible by 3.

Original entry on oeis.org

4, 8, 9, 11, 17, 21, 22, 24, 30, 34, 35, 37, 43, 47, 48, 50, 56, 60, 61, 63, 69, 73, 74, 76, 82, 86, 87, 89, 95, 99, 100, 102, 108, 112, 113, 115, 121, 125, 126, 128, 134, 138, 139, 141, 147, 151, 152, 154, 160, 164, 165, 167, 173, 177, 178, 180, 186, 190, 191

Offset: 1

Enrique Pérez Herrero, Sep 10 2013

nonn

a(n) appears to be congruent 4, 8, 9, 11 mod 13. - Ralf Stephan, Sep 12 2013

Wagstaff shows that N(p) = (p^p-1)/(p-1) is the period for all primes p < 102, for p=3 then N(3) = A054767(3) = 13, Bell numbers with indices less than or equal to 13 that are divisible by 3 are those with indices: 4, 8, 9, 11, so the conjecture holds. - Enrique Pérez Herrero, Sep 12 2013

Vaclav Kotesovec, Table of n, a(n) for n = 1..15384 (terms 1..1200 from Enrique Pérez Herrero)
J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423.
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, A016789, A155730, A054767.

Select[Range[1000], Mod[BellB[#],3] == 0&]

Conjectures from Colin Barker, Jul 16 2014: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(2*x^4+2*x^3+x^2+4*x+4) / ((x-1)^2*(x+1)*(x^2+1)). (End)

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 *)

A146101 Bell numbers (A000110) read mod 11.

Original entry on oeis.org

1, 1, 2, 5, 4, 8, 5, 8, 4, 5, 2, 2, 3, 7, 9, 1, 2, 2, 1, 9, 7, 4, 5, 10, 5, 10, 3, 4, 3, 10, 5, 0, 9, 4, 4, 4, 2, 7, 7, 2, 4, 5, 9, 2, 8, 8, 6, 9, 3, 9, 6, 9, 3, 0, 10, 5, 3, 4, 1, 1, 4, 4, 1, 3, 10, 4, 8, 7, 5, 2, 5, 8, 5, 4, 2, 3, 1, 4, 1, 7, 7, 2, 2, 9, 6, 5, 4, 5, 5, 8, 3, 9, 4, 0, 4, 0, 9, 9, 10, 2, 0, 1, 2

Offset: 0

N. J. A. Sloane, Feb 07 2009

nonn

Sequence is of period A054767(11) = 28531167061. - 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 28531167061.

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

Cf. A054767 (periods).

[Bell(n) mod 11: n in [0..100]]; // Vincenzo Librandi, Feb 01 2016

Mod[BellB[Range[0, 100]], 11] (* G. C. Greubel, Jan 31 2016 *)

A146102 Bell numbers (A000110) read mod 12.

Original entry on oeis.org

1, 1, 2, 5, 3, 4, 11, 1, 0, 3, 7, 6, 1, 1, 10, 5, 11, 0, 7, 5, 4, 3, 3, 10, 9, 1, 10, 1, 11, 8, 3, 1, 8, 7, 3, 6, 1, 9, 10, 1, 7, 8, 11, 9, 4, 11, 7, 6, 9, 1, 6, 1, 7, 4, 11, 5, 0, 7, 11, 10, 9, 9, 10, 9, 7, 4, 7, 5, 8, 3, 7, 2, 1, 9, 6, 1, 3, 4, 7, 1, 8, 11, 3, 10, 5, 1, 6, 9, 7, 0, 7, 1, 4, 11, 11, 6, 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, order 156.

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

Cf. A054767 (periods).

[Bell(n) mod 12: n in [0..100]]; // Vincenzo Librandi, Feb 01 2016

Mod[BellB[Range[0, 10000]], 12] (* G. C. Greubel, Jan 31 2016 *)

a(n+156) = a(n). - G. C. Greubel, Jan 31 2016 [See A054767. - Jianing Song, Jun 20 2025]

cp's OEIS Frontend

A155730 Indices of Bell numbers that are divisible by 5.

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Mathematica

A229004 Indices of Bell numbers divisible by 3.

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A146095 Bell numbers (A000110) read mod 5.

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Magma

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A146096 Bell numbers (A000110) read mod 6.

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Magma

Mathematica

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A146097 Bell numbers (A000110) read mod 7.

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Magma

Maple

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A146098 Bell numbers (A000110) read mod 8.

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Magma

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A146099 Bell numbers (A000110) read mod 9.

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Magma

Mathematica

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A146100 Bell numbers (A000110) read mod 10.

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Magma