A275949 Number of distinct nonzero digits that occur multiple times in factorial base representation of n: a(n) = A056170(A275735(n)).

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

Offset: 0

Author

Antti Karttunen, Aug 15 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 are no nonzero digits that are present multiple times, thus a(2) = 0.
For n=3 ("11") there is one distinct nonzero digit which occurs more than once, thus a(3) = 1.
For n=41 ("1221") there are two distinct nonzero digits ("1" and "2"), and both occur more than once, thus a(41) = 2.
For n=44 ( "1310") there are two distinct nonzero digits ("1" and "3"), but only the other (1) occurs more than once, thus a(44) = 1.
		

Links

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

Crossrefs

Cf. A056170, A275735.

Cf. A265349 (indices of zeros), A265350 (of terms > 0).

Cf. also A007623, A225901, A275806, A275947, A275948, A275964.

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++]; Count[Tally[Select[s, # > 0 &]][[;;, 2]], ?(# > 1 &)]]; Array[a, 100, 0] (* _Amiram Eldar, Feb 14 2024 *)

Python

from sympy import prime, factorint
from operator import mul
from functools import reduce
import collections
def a056170(n):
    f = factorint(n)
    return sum([1 for i in f if f[i]!=1])
def a007623(n, p=2): return n if n
Indranil Ghosh, Jun 20 2017

Scheme

(define (A275949 n) (A056170 (A275735 n)))

Formula

a(n) = A056170(A275735(n)).

Other identities and observations. For all n >= 0.

a(n) = A275947(A225901(n)).

A275806(n) = A275948(n) + a(n).

a(n) <= A275964(n).