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a(n) = smallest exponent k such that the string "1 2 ... n" appears in the decimal expansion of 2^k.
2

%I #19 Dec 09 2017 16:59:08

%S 0,7,81,283,684,1318,8792,15975,61274,314072,4057579

%N a(n) = smallest exponent k such that the string "1 2 ... n" appears in the decimal expansion of 2^k.

%C The first time "0" appears is in a(10). - _Robert G. Wilson v_, Feb 26 2013

%D Julian Havil, Impossible?: Surprising Solutions to Counterintuitive Conundrums, Princeton University Press 2008

%D Ross Honsberger, Ingenuity in mathematics, Random House/Singer School Division 1970

%e n=1: 2^0 = 1

%e n=2: 2^7 = 128

%e n=3: 2^81 = 2417851639229258349412352

%e n=4: 2^283 has 86 decimals, "1234" appears on decimals 68 - 71:

%e 2^283=

%e 15541351137805832567355695254588151253139254712417116170014499277911234281641667985408

%e n=5: 2^684 has 206 decimals, "12345" appears on decimals 99 - 103.

%t g[n_] := Block[{c = 0, k = 1}, While[k <= n, c = 10^Floor[1 + Log10[k]] c + k; k++]; c] (* from A007908 *); f[n_] := Block[{k = 0, s = ToString[g[n]]}, While[ StringPosition[ ToString[ 2^k], s] == {}, k++]; k]; Array[f, 10] (* _Robert G. Wilson v_, Feb 26 2013 *)

%Y Cf. A000079, A018802, A171132, A171242, A171489, A171652.

%K nonn,base

%O 1,2

%A Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 18 2009

%E a(6)-a(10) from _Robert G. Wilson v_, Feb 26 2013

%E a(11) from _Giovanni Resta_, Feb 26 2013