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A006889
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Exponent of least power of 2 having n consecutive 0's in its decimal representation.
(Formerly M4710)
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15
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0, 10, 53, 242, 377, 1491, 1492, 6801, 14007, 100823, 559940, 1148303, 4036338, 4036339, 53619497, 119476156, 146226201, 918583174, 4627233991, 11089076233
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listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.
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EXAMPLE
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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.
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MAPLE
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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;
fi
od:
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
import Data.List (elemIndex)
import Data.Maybe (fromJust)
a006889 = fromJust . (`elemIndex` a224782_list)
(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
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CROSSREFS
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KEYWORD
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nonn,hard,base,more
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AUTHOR
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P. D. Mitchelmore (dh115(AT)city.ac.uk)
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EXTENSIONS
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One more term from Sacha Roscoe (scarletmanuka(AT)iprimus.com.au), Dec 16 2002
a(17) from Sacha Roscoe (scarletmanuka(AT)iprimus.com.au), Feb 06 2007
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STATUS
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approved
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