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A276084
a(n) = Number of trailing zeros in primorial base representation of n (A049345); largest k such that A002110(k) divides n.
21
0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3
OFFSET
1,6
COMMENTS
Terms begin from a(1)=0 because for zero the count is ambiguous.
From Amiram Eldar, Mar 10 2021: (Start)
The asymptotic density of the occurrences of k is (prime(k+1)-1)/A002110(k+1).
The asymptotic mean of this sequence is Sum_{k>=1} 1/A002110(k) = 0.705230... (A064648). (End)
FORMULA
a(n) = A257993(n)-1.
Other identities. For all n >= 1:
A053589(n) = A002110(a(n)).
a(n) = A001221(A053589(n)) = A001221(A340346(n)). - Peter Munn, Jan 14 2021
EXAMPLE
For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), there are two trailing zeros, thus a(24) = 2.
MATHEMATICA
Table[If[# == 0, 0, j = #; While[! Divisible[n, Times @@ Prime@ Range@ j], j--]; j] &@ If[OddQ@ n, 0, k = 1; While[Times @@ Prime@ Range[k + 1] <= n, k++]; k], {n, 120}] (* or *)
nn = 120; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[Length@ TakeWhile[Reverse@ IntegerDigits[n, b], # == 0 &], {n, nn}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Length@ TakeWhile[Reverse@ f@ n, # == 0 &], {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)
PROG
(Scheme)
(define (A276084 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) (- i 1) (loop (/ (- n d) p) (+ 1 i))))))
(Python)
from sympy import nextprime, primepi
def a053669(n):
p = 2
while True:
if n%p!=0: return p
else: p=nextprime(p)
def a(n): return primepi(a053669(n)) - 1 # Indranil Ghosh, May 12 2017
CROSSREFS
One less than A257993.
Differs from the related A230403 for the first time at n=24.
Sequence in context: A097796 A117188 A341514 * A230403 A349907 A248908
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Aug 22 2016
STATUS
approved