A260736 a(0) = 0; for n >= 1, a(n) = A000035(n) + a(A257684(n)); in the factorial representation of n the number of digits with maximal possible value allowed in its location.

0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 0

Offset

0,6

Comments

In the factorial representation of n, given as {d_k, ..., d_3, d_2, d_1}, the maximal allowed digit for any position j is j. This sequence gives the number of digits in the whole representation [A007623(n)] that attain that maximum allowed value.

Links

Table of n, a(n) for n=0..120.

Index entries for sequences related to factorial base representation.

Formula

a(0) = 0; for n >= 1, a(n) = A000035(n) + a(A257684(n)).

Other identities. For all n >= 1:

a(n!-1) = n-1. [n!-1 also gives the first position where n-1 occurs.]

Example

For n=19, which has factorial representation "301", the digits at position 1 and 3, namely "1" and "3" are equal to their one-based position index, in other words, the maximal digits allowed in those positions (while "0" at position 2 is not), thus a(19) = 2.

Mathematica

a[n_] := Module[{k = n, m = 2, c = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r == m - 1, c++]; m++]; c]; Array[a, 100, 0] (* Amiram Eldar, Jan 23 2024 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A260736 n) (if (zero? n) 0 (+ (A000035 n) (A260736 (A257684 n)))))
(Python)
from sympy import factorial as f
def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
def a257684(n):
    x=str(a007623(n))[:-1]
    y="".join(str(int(i) - 1) if int(i)>0 else '0' for i in x)[::-1]
    return 0 if n==1 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
def a(n): return 0 if n==0 else n%2 + a(a257684(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

Crossrefs

Cf. A000035, A007623, A257684.

Cf. also A257511.

Sequence in context: A141747 A239706 A328616 * A342656 A293896 A066416

Adjacent sequences: A260733 A260734 A260735 * A260737 A260738 A260739

Keyword

nonn,base

Author

Antti Karttunen, Aug 27 2015

Status

approved