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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. 10

%I

%S 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,

%T 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,

%U 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

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

%C Terms begin from a(1)=0 because for zero the count is ambiguous.

%H Antti Karttunen, <a href="/A276084/b276084.txt">Table of n, a(n) for n = 1..2310</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(n) = A257993(n)-1.

%F Other identities. For all n >= 1:

%F A053589(n) = A002110(a(n)).

%e 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.

%t 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 *)

%t nn = 120; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[Length@ TakeWhile[Reverse@ IntegerDigits[n, b], # == 0 &], {n, nn}] (* Version 10.2, or *)

%t 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 *)

%o (Scheme)

%o (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))))))

%o (Python)

%o from sympy import nextprime, primepi

%o def a053669(n):

%o p = 2

%o while True:

%o if n%p!=0: return p

%o else: p=nextprime(p)

%o def a(n): return primepi(a053669(n)) - 1 # _Indranil Ghosh_, May 12 2017

%Y Cf. A000040, A002110, A049345, A053589.

%Y One less than A257993.

%Y Differs from the related A230403 for the first time at n=24.

%K nonn,base

%O 1,6

%A _Antti Karttunen_, Aug 22 2016

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Last modified May 24 18:34 EDT 2019. Contains 323534 sequences. (Running on oeis4.)