A276084 - OEIS

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

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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