login
A275949
Number of distinct nonzero digits that occur multiple times in factorial base representation of n: a(n) = A056170(A275735(n)).
5
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
0,42
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).
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 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.
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 *)
PROG
(Scheme) (define (A275949 n) (A056170 (A275735 n)))
(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<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 a056170(a275735(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 20 2017
CROSSREFS
Cf. A265349 (indices of zeros), A265350 (of terms > 0).
Sequence in context: A116929 A059984 A046675 * A357924 A351563 A362983
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 15 2016
STATUS
approved