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

0,6

Links

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

Index entries for sequences related to factorial base representation.

Formula

a(n) = 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).

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 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.

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 *)

Prog

(Scheme) (define (A275948 n) (A056169 (A275735 n)))
(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<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 a056169(a275735(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 20 2017

Crossrefs

Cf. A056169, A275735.

Cf. also A060130, A225901, A275806, A275946, A275949, A275964.

Sequence in context: A070176 A092606 A374133 * A356325 A073253 A004198

Adjacent sequences: A275945 A275946 A275947 * A275949 A275950 A275951

Keyword

nonn,base

Author

Antti Karttunen, Aug 15 2016

Status

approved