A276084 a(n) = Number of trailing zeros in primorial base representation of n (A049345); largest k such that A002110(k) divides n.

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)

Links

Antti Karttunen, Table of n, a(n) for n = 1..2310

Index entries for sequences related to primorial base

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

Cf. A000040, A001221, A002110, A049345, A053589, A064648, A340346.

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

Adjacent sequences: A276081 A276082 A276083 * A276085 A276086 A276087

Keyword

nonn,base

Author

Antti Karttunen, Aug 22 2016

Status

approved