A230403 a(n) = the largest k such that (k+1)! divides n; the number of trailing zeros in the factorial base representation of n (A007623(n)).

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1

Offset

1,6

Comments

Many of the comments given in A055881 apply also here.

From Amiram Eldar, Mar 10 2021: (Start)

The asymptotic density of the occurrences of k is (k+1)/(k+2)!.

The asymptotic mean of this sequence is e - 2 = 0.718281... (A001113 - 2). (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..10080

Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.

Index entries for sequences related to factorial base representation

Formula

a(n) = A055881(n)-1.

Example

In factorial number base representation (A007623), the numbers from 1 to 9 are represented as:
n  A007623(n)  a(n) (gives the number of trailing zeros)
1        1       0
2       10       1
3       11       0
4       20       1
5       21       0
6      100       2
7      101       0
8      110       1
9      111       0

Mathematica

With[{b = MixedRadix[Range[12, 2, -1]]}, Array[LengthWhile[Reverse@ IntegerDigits[#, b], # == 0 &] &, 105]] (* Michael De Vlieger, Jun 03 2020 *)

Prog

(Scheme)
(define (A230403 n) (if (zero? n) 0 (let loop ((n n) (i 2)) (cond ((not (zero? (modulo n i))) (- i 2)) (else (loop (/ n i) (1+ i)))))))

Crossrefs

Cf. A001113, A055881. Bisection: A230404.

A few sequences related to factorial base representation (A007623): A034968, A084558, A099563, A060130, A227130, A227132, A227148, A227149, A153880.

Analogous sequence for binary system: A007814.

Sequence in context: A117188 A341514 A276084 * A349907 A248908 A133565

Adjacent sequences: A230400 A230401 A230402 * A230404 A230405 A230406

Keyword

nonn,base,easy

Author

Antti Karttunen, Oct 31 2013

Status

approved