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

A033107 Number of 7's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A033093, A033095, A033097, A033099, A033101, A033103, A033105, A033108, A033109, A033111.

Programs

  • Maple
    f:= proc(n) local b;`+`(seq(numboccur(7,convert(n,base,b)),b=8..n+1)) end proc:
map(f, [$1..100]); # Robert Israel, Aug 20 2024
  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 7]; Array[f, 90] (* Robert G. Wilson v, Nov 14 2012 *)

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 *)

A033097 Number of 2's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 2, 4, 2, 4, 4, 4, 3, 7, 4, 6, 7, 7, 5, 9, 5, 9, 8, 9, 8, 14, 7, 7, 8, 9, 6, 13, 8, 12, 12, 10, 11, 16, 9, 11, 12, 16, 11, 16, 10, 14, 15, 12, 11, 20, 14, 17, 16, 15, 12, 19, 15, 19, 16, 16, 15, 25, 15, 17, 19, 19, 17, 21, 16, 19, 18
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033098, A033099, A033101, A033103, A033105, A033107, A033109, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 2]; Array[f, 70] (* Robert G. Wilson v, Nov 14 2012 *)

A033099 Number of 3's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 2, 5, 2, 3, 4, 5, 2, 5, 3, 6, 4, 4, 3, 8, 5, 5, 6, 8, 3, 8, 4, 7, 5, 5, 6, 10, 5, 6, 7, 11, 6, 10, 6, 9, 9, 7, 6, 13, 7, 9, 9, 11, 7, 12, 10, 13, 9, 9, 8, 17, 6, 8, 11, 12, 9, 12, 8, 11, 9, 13, 8, 18, 10, 10, 13, 13, 10
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033097, A033100, A033101, A033103, A033105, A033107, A033109, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 3]; Array[f, 80] (* Robert G. Wilson v, Nov 14 2012 *)

A033101 Number of 4's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033097, A033099, A033102, A033103, A033105, A033107, A033109, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 4]; Array[f, 88] (* Robert G. Wilson v, Nov 14 2012 *)

A033103 Number of 5's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033097, A033099, A033101, A033104, A033105, A033107, A033109, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 5]; Array[f, 90] (* Robert G. Wilson v, Nov 14 2012 *)

A033105 Number of 6's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033097, A033099, A033101, A033103, A033106, A033107, A033109, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 6]; Array[f, 90] (* Robert G. Wilson v, Nov 14 2012 *)

A033109 Number of 8's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 6, 1, 2, 3, 4, 3, 5, 2, 4, 3, 5, 2, 8, 2, 3, 4, 4, 3, 5, 2, 7, 5, 3
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033097, A033099, A033101, A033103, A033105, A033107, A033110, A033111.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 8]; Array[f, 90] (* Robert G. Wilson v, Nov 14 2012 *)

A033111 Number of 9's when n is written in base b for 2<=b<=n+1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A033093, A033095, A033097, A033099, A033101, A033103, A033105, A033107, A033109, A033112.

Programs

  • Mathematica
    f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 9]; Array[f, 90](* Robert G. Wilson v, Nov 14 2012 *)
Showing 1-10 of 10 results.

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