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

0,6

Links

Antti Karttunen, Table of n, a(n) for n = 0..40320

Index entries for sequences related to factorial base representation.

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.

Example

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.

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 *)

Prog

(Scheme) (define (A275806 n) (A001221 (A275735 n)))
(Python)
from sympy import prime, primefactors
from operator import mul
import collections
def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
def a275735(n):
    y=collections.Counter(map(int, list(str(a007623(n)).replace("0", "")))).most_common()
    return 1 if n==0 else reduce(mul, [prime(y[i][0])**y[i][1] for i in range(len(y))])
def a(n): return len(primefactors(a275735(n)))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 20 2017

Crossrefs

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

Cf. also A153880, A255411.

Sequence in context: A254011 A361919 A002635 * A362833 A228369 A296773

Adjacent sequences: A275803 A275804 A275805 * A275807 A275808 A275809

Keyword

nonn,base

Author

Antti Karttunen, Aug 11 2016

Status

approved