A275806 a(n) = number of distinct nonzero digits in factorial base representation of n.

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

Offset: 0

Author

Antti Karttunen, Aug 11 2016

Keywords

• nonn
• base

Examples

			For n=0, with factorial base representation (A007623) also 0, there are no nonzero digits, thus a(0) = 0.
For n=2, with factorial base representation "10", there is one distinct nonzero digit, thus a(2) = 1.
For n=3, with factorial base representation "11", there is just one distinct nonzero digit, thus a(3) = 1.
For n=44, with factorial base representation "1310", there are two distinct nonzero digits ("1" and "3"), thus a(44) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..40320
• Index entries for sequences related to factorial base representation.

Crossrefs

Cf. A000142, A001221, A007623, A033312, A060130, A060502, A225901, A265349, A275735.

Cf. also A153880, A255411.

Programs

Mathematica

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Length[Union[Select[s, # > 0 &]]]]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

Python

from sympy import prime, primefactors
from operator import mul
import collections
def a007623(n, p=2): return n if n
Indranil Ghosh, Jun 20 2017

Scheme

(define (A275806 n) (A001221 (A275735 n)))

Formula

a(n) = A001221(A275735(n)).

a(n) = A060502(A225901(n)).

Other identities. For all n >= 0:

a(n) = a(A153880(n)) = a(A255411(n)). [The shift-operations do not change the number of distinct nonzero digits.]

a(A265349(n)) = A060130(A265349(n)).

a(A000142(n)) = 1.

a(A033312(n)) = n-1, for all n >= 1.