cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A033093 Number of 0's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 1, 5, 5, 5, 3, 6, 2, 3, 3, 8, 5, 9, 5, 8, 5, 4, 2, 9, 5, 5, 7, 9, 5, 8, 2, 11, 9, 8, 8, 13, 6, 7, 6, 11, 5, 9, 3, 7, 8, 5, 3, 13, 7, 10, 8, 9, 5, 12, 7, 11, 6, 5, 3, 13, 3, 4, 6, 15, 12, 14, 8, 11, 9, 12, 6, 18, 8, 9, 11, 11, 9, 11, 5, 14, 13
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A033094, A033095, A033097, A033099, A033101, A033103, A033105, A033107, A033109, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 0]; Array[f, 90] (* Robert G. Wilson v, Nov 14 2012 *)

Formula

Conjecture: lim inf a(n) = infinity. The lim inf grows quite slowly: e.g. a(2047)=7. Weaker conjecture: 2, 3 and 7 are the only n for which a(n) = 1. Note that a(n)=1 implies n=2 or n=2^k-1; probabilistically, the chance that 2^k-1 has no zeros just in base 3 is C^k, where C = (2/3)^(log(2)/log(3)) ~ .774, so the number of exceptions should be finite. It appears that 2^15-1 is the last n with no zeros in either base 2 or 3 (and it does have zeros in base 5). - Franklin T. Adams-Watters, Nov 07 2005
G.f.: (Sum_{b>=2} (Sum_{k>=0} x^(b^(k+1))/(Sum_{0<=iFranklin T. Adams-Watters, Nov 07 2005

A033096 Number of 1's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

1, 2, 5, 9, 15, 21, 30, 36, 46, 56, 68, 79, 95, 108, 123, 137, 153, 166, 184, 199, 220, 240, 261, 277, 301, 321, 344, 367, 393, 418, 450, 472, 498, 523, 548, 576, 610, 638, 670, 700, 735, 765, 802, 833, 868, 904, 939, 970, 1011, 1045
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033095, A033098, A033100, A033102, A033104, A033106, A033108, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 1] - (n - 1); Array[f, 50] (* Robert G. Wilson v, Nov 14 2012 *)

A033098 Number of 2's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 12, 16, 20, 24, 27, 34, 38, 44, 51, 58, 63, 72, 77, 86, 94, 103, 111, 125, 132, 139, 147, 156, 162, 175, 183, 195, 207, 217, 228, 244, 253, 264, 276, 292, 303, 319, 329, 343, 358, 370, 381, 401, 415, 432, 448
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033097, A033100, A033102, A033104, A033106, A033108, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 2] - (n - 2); Array[f, 53] (* Robert G. Wilson v, Nov 14 2012 *)

A033100 Number of 3's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 14, 19, 21, 24, 28, 33, 35, 40, 43, 49, 53, 57, 60, 68, 73, 78, 84, 92, 95, 103, 107, 114, 119, 124, 130, 140, 145, 151, 158, 169, 175, 185, 191, 200, 209, 216, 222, 235, 242, 251, 260, 271, 278
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033099, A033102, A033104, A033106, A033108, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 3] - (n - 3); f[1] = f[2] = 0; Array[f, 56] (* Robert G. Wilson v, Nov 14 2012 *)

A033102 Number of 4's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 16, 19, 21, 25, 27, 32, 35, 38, 40, 46, 50, 53, 56, 61, 64, 71, 73, 78, 82, 86, 91, 98, 101, 105, 109, 117, 121, 129, 133, 140, 148, 152, 155, 164, 169, 176, 180, 186, 190, 198, 204
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033100, A033101, A033104, A033106, A033108, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 4] - (n - 4); f[1] = f[2] = f[3] = 0; Array[f, 59] (* Robert G. Wilson v, Nov 14 2012 *)

A033104 Number of 5's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 16, 19, 20, 22, 24, 26, 27, 31, 33, 36, 39, 43, 45, 51, 53, 57, 60, 63, 67, 74, 76, 79, 82, 88, 91, 98, 100, 104, 109, 113, 116, 124, 127, 132, 136, 141, 144, 151, 155, 162, 166
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033100, A033102, A033103, A033106, A033108, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 5] - (n - 5); f[1] = f[2] = f[3] = f[4] = 0; Array[f, 62] (* Robert G. Wilson v, Nov 14 2012 *)

A033106 Number of 6's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 21, 23, 24, 27, 28, 30, 32, 35, 36, 39, 40, 43, 45, 47, 49, 54, 56, 59, 62, 67, 69, 75, 77, 81, 85, 88, 90, 97, 101, 105, 108, 112, 114, 120, 124, 131, 134, 137
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033100, A033102, A033104, A033105, A033108, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 6] - (n - 6); f[1] = f[2] = f[3] = f[4] = f[5] = 0; Array[f, 64] (* Robert G. Wilson v, Nov 14 2012 *)

A033108 Number of 7's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 24, 25, 27, 29, 31, 32, 35, 36, 39, 41, 43, 44, 48, 49, 51, 53, 57, 58, 61, 62, 65, 68, 70, 71, 76, 78, 82, 85, 89, 91, 96, 99, 105, 108, 111, 113, 120
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033100, A033102, A033104, A033106, A033107, A033110, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 7] - (n - 7); f[1] = f[2] = f[3] = f[4] = f[5] = f[6] = 0; Array[f, 67] (* Robert G. Wilson v, Nov 14 2012 *)

A033110 Number of 8's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 26, 28, 29, 32, 33, 35, 37, 39, 40, 44, 45, 47, 49, 52, 53, 56, 57, 60, 63, 65, 66, 70, 71, 74, 76, 79, 80, 84, 86, 89, 91, 93, 94, 100
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033100, A033102, A033104, A033106, A033108, A033109, A033112.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 8] - (n - 8); f[1] = f[2] = f[3] = f[4] = f[5] = f[6] = f[7] = 0; Array[f, 68] (* Robert G. Wilson v, Nov 14 2012 *)

A033112 Number of 9's when k is written in base b for all b and k satisfying 2<=b<=n+1, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 29, 30, 32, 34, 36, 37, 40, 41, 43, 45, 48, 49, 52, 53, 56, 58, 60, 61, 65, 66, 69, 71, 74, 75, 78, 80, 83, 85, 87, 88, 94
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033094, A033096, A033098, A033100, A033102, A033104, A033106, A033108, A033110, A033111.

Programs

  • Mathematica
    f[n_] := Count[ Flatten@ Table[ IntegerDigits[k, b], {k, n}, {b, 2, n + 1}], 9] - (n - 9); f[1] = f[2] = f[3] = f[4] = f[5] = f[6] = f[7] = f[8] = 0; Array[f, 69] (* Robert G. Wilson v, Nov 14 2012 *)
Showing 1-10 of 10 results.

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