A275948 Number of nonzero digits that occur only once in factorial base representation of n: a(n) = A056169(A275735(n)).

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

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 is one nonzero digit, thus a(2) = 1.
For n=3 (= "11") there is no nonzero digit which would occur just once, thus a(3) = 0.
For n=23 (= "321") there are three nonzero digits and each of those digits occurs just once, thus a(23) = 3.
For n=44 (= "1310") there are two distinct nonzero digits ("1" and "3"), but only the other (3) occurs just 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. A056169, A275735.

Cf. also A060130, A225901, A275806, A275946, A275949, 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 07 2024 *)

Python

from sympy import prime, factorint
from operator import mul
from functools import reduce
import collections
def a056169(n):
    f=factorint(n)
    return 0 if n==1 else 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 (A275948 n) (A056169 (A275735 n)))

Formula

a(n) = A056169(A275735(n)).

Other identities. For all n >= 0.

a(n) = A275946(A225901(n)).

A275806(n) = a(n) + A275949(n).

A060130(n) = a(n) + A275964(n).