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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. 14
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 A248908 A133565 Adjacent sequences:  A276081 A276082 A276083 * A276085 A276086 A276087 KEYWORD nonn,base AUTHOR Antti Karttunen, Aug 22 2016 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)