A215884 -id:A215884

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A215879

Written in base 3, n ends in a(n) consecutive nonzero digits.

+10
7

0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 1, 1, 0, 3, 3, 0, 3, 3, 0, 1, 1, 0, 3, 3, 0, 3, 3, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 1, 1, 0, 4, 4, 0, 4, 4, 0, 1, 1, 0, 4, 4, 0, 4, 4, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 1, 1, 0, 4, 4, 0, 4, 4, 0, 1, 1, 0, 4, 4, 0, 4, 4, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 1, 1, 0, 3, 3, 0, 3, 3, 0, 1, 1, 0, 3, 3, 0, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Somehow complementary to A007949, the 3-adic valuation of n.

The base 2 analog of this sequence essentially coincides with the 2-adic valuation A007814 (up to a shift in the index).

One gets back the same sequence by concatenation of the pattern (0,1,1) successively multiplied by a(n)+1 = 1, 2, 2, 1, 3, 3, ... for n = 0, 1, 2, 3, 4, 5, .... This is equivalent to the formula (a(n)+1)*(0, 1, 1) = a(3n, 3n+1, 3n+2). - M. F. Hasler, Aug 26 2012, corrected Aug 23 2022

a(A008585(n)) = 0; a(A001651(n)) > 0. - Reinhard Zumkeller, Dec 28 2012

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

FORMULA

a(3^(t+1)*k+m) = t for 3^t > m > 3^(t-1).

a(3n) = 0, a(3n+1) = a(3n+2) = a(n)+1. - M. F. Hasler, Aug 26 2012, corrected thanks to a remark from Jianing Song, Aug 23 2022

EXAMPLE

The numbers 0, 1, 2, 3, 4, 5, 6, 7 are written in base 3 as 0, 1, 2, 10, 11, 12, 20, 21 and thus end in a(0..7) = 0, 1, 1, 0, 2, 2, 0, 2 nonzero digits.

MATHEMATICA

cnzd[n_]:=Module[{idn3=IntegerDigits[n, 3], len}, len=Length[idn3]; Which[ idn3[[len]] == 0, 0, Position[idn3, 0]=={}, len, True, len-Position[idn3, 0] [[-1, 1]]]]; Array[cnzd, 110, 0] (* Harvey P. Dale, Jun 07 2016 *)

PROG

(PARI) A215879(n, b=3)=n=divrem(n, b); for(c=0, oo, n[2]||return(c); n=divrem(n[1], b))

(PARI) a(n)=my(k); while(n%3, n\=3; k++); k \\ Charles R Greathouse IV, Sep 26 2013

(Haskell)

a215879 n = if t == 0 then 0 else a215879 n' + 1

where (n', t) = divMod n 3

-- Reinhard Zumkeller, Dec 28 2012

(Python)

def A215879(n):

c = 0

while (a:=divmod(n, 3))[1]:

c += 1

n = a[0]

return c # Chai Wah Wu, Oct 15 2022

CROSSREFS

The base-4, base-5 and base-10 analogs of this sequence are given in A215883, A215884 and A215887.

Cf. A007089.

KEYWORD

nonn,base,nice

AUTHOR

M. F. Hasler, Aug 25 2012

STATUS

approved

A215887

Written in decimal, n ends in a(n) consecutive nonzero digits.

+10
6

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,12

COMMENTS

Sequences A215879, A215883 and A215884 are the base 3, 4 and 5 analogs, while the base 2 analog of this sequence coincides (up to a shift in the index) with the 2-adic valuation A007814, see comments there.

Starting indexing with k=0 for the rightmost digit, a(n) gives the index of the least significant zero in the decimal representation of n. This may also be the index of the leading zero if there are no zeros in the number itself (A052382). - Antti Karttunen, Dec 07 2017

First occurrence of k is A002275(k). - Robert G. Wilson v, Dec 07 2017

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..11111

EXAMPLE

Numbers which are multiples of 10 have no nonzero digit at their (right) end, thus a(10*k) = 0.

If numbers are congruent to 1,...,9 mod 100, then they end in a nonzero digit, but do not have more than 1 concatenated nonzero digits at their right end: Thus, a(100k+m)=1 for 0 < m < 10.

In the same way, a(k*10^(e+1)+m) = e if 10^e > m > 10^(e-1).

MATHEMATICA

Table[Which[Divisible[n, 10], 0, FreeQ[IntegerDigits[n], 0], IntegerLength[ n], True, Position[ Reverse[ IntegerDigits[n]], 0]-1], {n, 0, 110}] // Flatten (* Harvey P. Dale, Sep 05 2017 *)

f[n_] := Block[{c = 0, m = n}, While[Mod[m, 10] > 0, m = Floor[m/10]; c++]; c]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 07 2017 *)

PROG

(PARI) a(n, b=10)= n=divrem(n, b); for(c=0, 9e9, n[2] || return(c); n=divrem(n[1], b))

(PARI) a(n)=my(k); while(n%10, n\=10; k++); k \\ Charles R Greathouse IV, Sep 26 2013

CROSSREFS

Cf. A002275, A052382, A339012 (factorial base).

KEYWORD

nonn,base,easy

AUTHOR

M. F. Hasler, Aug 25 2012

EXTENSIONS

More terms from Antti Karttunen, Dec 07 2017

STATUS

approved

A215883

When written in base 4, n ends in a(n) consecutive nonzero digits.

+10
3

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 1, 1, 0, 3, 3, 3, 0, 3, 3, 3, 0, 3, 3, 3, 0, 1, 1, 1, 0, 3, 3, 3, 0, 3, 3, 3, 0, 3, 3, 3, 0, 1, 1, 1, 0, 3, 3, 3, 0, 3, 3, 3, 0, 3, 3, 3, 0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 1, 1, 0, 4, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

Sequences A215879, A215884 and A215887 are the base 3, 5 and 10 analog, while the base 2 analog of this sequence coincides (up to a shift in the index) with the 2-adic valuation A007814, see comments there.

LINKS

Table of n, a(n) for n=0..86.

FORMULA

a(4^(t+1)*k+m) = t for 4^t > m > 4^(t-1).

EXAMPLE

The numbers 0,1,2,3,4,5,6,7 are written in base 4 as 0,1,2,3,10,11,12,13 and thus end in a(0..7)=0,1,1,1,0,2,2,2 nonzero digits.

PROG

(PARI) a(n, b=4)=n=divrem(n, b); for(c=0, 9e9, n[2]||return(c); n=divrem(n[1], b))

(PARI) a(n)=my(k); while(n%4, n>>=2; k++); k \\ Charles R Greathouse IV, Sep 26 2013

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Aug 25 2012

STATUS

approved

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