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 1-4 of 4 results.

A071156 Apart from the initial term (0), lists all integers whose factorial expansion ends with 1 (i.e., are odd numbers), do not contain a digit zero and each successive digit to the left is at most one greater than the preceding digit.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 23, 33, 35, 39, 41, 47, 57, 59, 63, 65, 71, 87, 89, 95, 119, 153, 155, 159, 161, 167, 177, 179, 183, 185, 191, 207, 209, 215, 239, 273, 275, 279, 281, 287, 297, 299, 303, 305, 311, 327, 329, 335, 359, 417, 419, 423, 425, 431, 447, 449
Offset: 0

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Author

Antti Karttunen, May 14 2002

Keywords

Comments

a(n) = A085198(A014486(n)) = A071155(A057164(n)). Catalan numbers A000108(n) gives the number of terms whose factorial expansion contain n digits.
0 is included by considering it to have the empty string as its factorial base representation. - Franklin T. Adams-Watters, Jun 28 2006

Crossrefs

The beginning of this sequence expanded in the factorial number system: A071158. Inverse function: A085199. First differences: A085191.
Cf. A000108 (row lengths), A071155, A120696.

A120695 Set partitions reversed interpreted as factorial base numbers.

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 11, 17, 23, 33, 57, 39, 63, 87, 35, 59, 83, 41, 65, 89, 47, 71, 95, 119, 153, 273, 177, 297, 417, 159, 279, 399, 183, 303, 423, 207, 327, 447, 567, 155, 275, 395, 179, 299, 419, 203, 323, 443, 563, 161, 281, 401, 185, 305, 425, 209, 329, 449
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jun 28 2006

Keywords

Examples

			The third set partition is {1,2}, 21 in base factorial is 5, so a(3) = 5.
Triangle begins:
   0;
   1;
   3,  5;
   9, 15, 11, 17, 23;
  33, 57, 39, 63, 87, 35, 59, 83, 41, 65, 89, 47, 71, 95, 119;

Links

Crossrefs

Cf. A000110 (row lengths), A120698, A120696 (sorted), A120697, A071155.
Column k=0 gives A007489.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, [0], [seq(map(
       x-> x+j*t!, b(n-1, max(m, j), t+1))[], j=1..m+1)])
    end:
T:= n-> b(n, 0, 1)[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Apr 04 2016

A382262 Nonnegative numbers whose factorial base expansion, when read from right to left, corresponds to the ordinal transform of some finite sequence, with offset 0.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 23, 33, 35, 39, 47, 57, 59, 63, 83, 87, 119, 153, 155, 159, 167, 177, 179, 183, 203, 207, 239, 273, 275, 279, 287, 297, 323, 327, 395, 399, 417, 419, 423, 527, 563, 567, 719, 873, 875, 879, 887, 897, 899, 903, 923, 927, 959, 993, 995
Offset: 0

Views

Author

Rémy Sigrist, Mar 19 2025

Keywords

Comments

We ignore leading zeros in factorial base expansions.
A subsequence of A120696.
There are A000085(k) terms whose factorial base expansion has k digits.

Examples

			The first terms, in decimal and in factorial base, are:
  n   a(n)  fact(a(n))
  --  ----  ----------
   0     0  0
   1     1  1
   2     3  1,1
   3     5  2,1
   4     9  1,1,1
   5    11  1,2,1
   6    15  2,1,1
   7    23  3,2,1
   8    33  1,1,1,1
   9    35  1,1,2,1
  10    39  1,2,1,1
  11    47  1,3,2,1
  12    57  2,1,1,1
  13    59  2,1,2,1
  14    63  2,2,1,1
  15    83  3,1,2,1
  16    87  3,2,1,1
  17   119  4,3,2,1
  18   153  1,1,1,1,1

Links

Crossrefs

Cf. A000085, A120696, A382263.

Programs

  • PARI
    \\ See Links section.

A382313 The factorial base expansion of a(n) corresponds to the restricted growth sequence of that of n (when read from right to left).

Original entry on oeis.org

0, 1, 5, 3, 5, 5, 15, 11, 17, 9, 23, 11, 15, 23, 23, 15, 17, 17, 15, 23, 23, 15, 23, 23, 57, 41, 59, 39, 83, 47, 63, 35, 65, 33, 95, 35, 87, 47, 71, 39, 89, 41, 87, 47, 71, 39, 119, 47, 57, 89, 83, 87, 59, 71, 87, 83, 89, 57, 71, 59, 63, 95, 95, 63, 65, 65, 87
Offset: 0

Views

Author

Rémy Sigrist, Mar 21 2025

Keywords

Comments

We ignore nonleading zeros in factorial base expansions.
All terms belong to A120696.

Examples

			The first terms, in decimal and in factorial base, are:
  n   a(n)  fact(n)  fact(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     1        1           1
   2     5       10          21
   3     3       11          11
   4     5       20          21
   5     5       21          21
   6    15      100         211
   7    11      101         121
   8    17      110         221
   9     9      111         111
  10    23      120         321
  11    11      121         121
  12    15      200         211
  13    23      201         321
  14    23      210         321
  15    15      211         211

Links

Crossrefs

See A382269 for a similar sequence.
Cf. A120696.

Programs

  • PARI
    a(n) = { my (v = 0, m = Map(), u = 0, d, c); for (r = 2, oo, if (n==0, return (v), d = n%r; n \= r; if (!mapisdefined(m, d, &c), mapput(m, d, c=u++);); v += c*(r-1)!;);); }

Formula

a(a(n)) = a(n).
Showing 1-4 of 4 results.

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