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 1-3 of 3 results.

A208575 Product of digits of n in factorial base.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 8, 0, 0, 0, 6, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 6, 0, 12, 0, 0, 0, 9, 0, 18
Offset: 0

Views

Author

Charles R Greathouse IV, Feb 28 2012

Keywords

Links

Crossrefs

Cf. A007623, A007954, A031346, A208576, A208277, A227153, A227154, A286590.

Programs

  • Mathematica
    (* For the definition of the factorial base version of IntegerDigits, see A007623 *) Table[Times@@factBaseIntDs[n], {n, 0, 99}] (* Alonso del Arte, Feb 28 2012 *)
  • PARI
    a(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s
  • Python
    from functools import reduce
from operator import mul
def A(n, p=2):
    return n if n
    Indranil Ghosh, Jun 19 2017

A208576 Multiplicative persistence of n in factorial base.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2
Offset: 0

Views

Author

Charles R Greathouse IV, Feb 28 2012

Keywords

Comments

Diamond and Reidpath prove that a(2n) = 1 for n > 0, a(n) = 2 if n is contains an even digit but no 0's in its factorial base representation. If a(n) > 2 then 3 | n.
Further modular properties can be easily proved. For example, a(n) > 2 implies that n is 33, 45, 81, or 93 mod 120.

Links

Crossrefs

Cf. A007623, A031346, A208575, A208277.

Programs

  • PARI
    pr(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s
a(n)=my(t);while(n>1,t++;n=pr(n));t

Formula

a(0) = a(1) = 0; for n > 1, a(n) = 1 + a(A208575(n)). - Antti Karttunen, Nov 14 2018

A263130 Least number such that the product of its digits in factorial base is n.

Original entry on oeis.org

1, 5, 21, 17, 633, 23, 36153, 65, 93, 635, 443122713, 71, 81474226713, 36155, 645, 113, 6069010670156313, 95, 2318037293294156313, 641, 36165, 443122715, 595774037991797891660313, 119, 4233, 81474226715, 453, 36161, 256727294482662730300616548940313, 647
Offset: 1

Views

Author

Paul Tek, Oct 10 2015

Keywords

Comments

The product of digits in factorial base is given by A208575.
All terms are odd.
Each prime number sets a new record.
a(p) = p*(p!) + Sum_{k=1..p-1} k! for any prime p.
a(n!) = A033312(n+1) for any n>0.
A208576(a(n)) = A208576(n)+1 for any n>1.

Examples

			The first terms of the sequence are:
+----+-------------+----------------------------+
| n  | a(n)        | a(n) in factorial base     |
+----+-------------+----------------------------+
|  1 |           1 |                          1 |
|  2 |           5 |                        2_1 |
|  3 |          21 |                      3_1_1 |
|  4 |          17 |                      2_2_1 |
|  5 |         633 |                  5_1_1_1_1 |
|  6 |          23 |                      3_2_1 |
|  7 |       36153 |              7_1_1_1_1_1_1 |
|  8 |          65 |                    2_2_2_1 |
|  9 |          93 |                    3_3_1_1 |
| 10 |         635 |                  5_1_1_2_1 |
| 11 |   443122713 |     11_1_1_1_1_1_1_1_1_1_1 |
| 12 |          71 |                    2_3_2_1 |
| 13 | 81474226713 | 13_1_1_1_1_1_1_1_1_1_1_1_1 |
| 14 |       36155 |              7_1_1_1_1_2_1 |
| 15 |         645 |                  5_1_3_1_1 |
| 16 |         113 |                    4_2_2_1 |
+----+-------------+----------------------------+

Links

Crossrefs

Cf. A033312, A200748, A208277, A208575, A208576.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors@ n, g, k, m = {1}}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; Do[k = Max@ Select[d, # <= i &]; If[! IntegerQ@ k, AppendTo[m, 1], d = Divisors[Last[d]/k]; AppendTo[m, k]]; If[d == {1}, Break[]], {i, 2, n}]; Reverse@ m]; Table[FromDigits[#, MixedRadix[Reverse@ Range[2, Length@ #]]] &@ f@ n, {n, 30}] (* Michael De Vlieger, Oct 12 2015, Version 10.2 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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