A006889 - OEIS

A006889

Exponent of least power of 2 having n consecutive 0's in its decimal representation.
(Formerly M4710)

0, 10, 53, 242, 377, 1491, 1492, 6801, 14007, 100823, 559940, 1148303, 4036338, 4036339, 53619497, 119476156, 146226201, 918583174, 4627233991, 11089076233

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

OFFSET

0,2

COMMENTS

A224782(a(n)) = n and A224782(m) <> n for m < a(n). - Reinhard Zumkeller, Apr 30 2013

The name of this sequence previously began "Least power of 2 having exactly n consecutive 0's...". The word "exactly" was unnecessary because the least power of 2 having at least n consecutive 0's in its decimal representation will always have exactly n consecutive 0's. The previous power of two will have had n-1 consecutive 0's with a "5" immediately to the left. - Clive Tooth, Jan 22 2016

a(20) is greater than 12*10^9. - Benjamin Chaffin, Jan 18 2017

REFERENCES

Julian Havil, Impossible?: Surprising Solutions to Counterintuitive Conundrums, Princeton University Press 2008, chapter 15, p. 176ff

Popular Computing (Calabasas, CA), Zeros in Powers of 2, Vol. 3 (No. 25, Apr 1975), page PC25-16 [Gives a(1)-a(8)]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..19.

E. Karst and U. Karst, The first power of 2 with 8 consecutive zeros, Math. Comp. 18 (1964), 508-508.

Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

EXAMPLE

2^53619497 is the smallest power of 2 to contain a run of 14 consecutive zeros in its decimal form.

2^119476156 (a 35965907-digit number) contains the sequence ...40030000000000000008341... about one third of the way through.

2^4627233991 (a 1392936229-digit number) contains the sequence "813000000000000000000538" about 99.5% of the way through. The computation took about six months.

MAPLE

A[0]:= 0:

m:= 1:

for n from 1 while m <= 9 do

S:= convert(2^n, string);

if StringTools:-Search(StringTools:-Fill("0", m), S) <> 0 then

A[m]:= n;

m:= m+1;

od:

seq(A[i], i=0..9); # Robert Israel, Jan 22 2016

MATHEMATICA

a = ""; Do[ a = StringJoin[a, "0"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {}, k++ ]; Print[k], {n, 1, 10} ] (* Robert G. Wilson v, edited by Clive Tooth, Jan 25 2016 *)

PROG

(Haskell)

import Data.List (elemIndex)

import Data.Maybe (fromJust)

a006889 = fromJust . (`elemIndex` a224782_list)

-- Reinhard Zumkeller, Apr 30 2013

(PARI) conseczerorec(n) = my(d=digits(n), i=0, r=0, x=#Str(n)); while(x > 0, while(d[x]==0, i++; x--); if(i > r, r=i); i=0; x--); r

a(n) = my(k=0); while(conseczerorec(2^k) < n, k++); k \\ Felix Fröhlich, Sep 27 2016

CROSSREFS

Cf. A031146.

Cf. also A131535, A131536, A259089, A063565, A259091, A259092.

Sequence in context: A216938 A298564 A063899 * A300422 A300684 A300607