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Number of decimal digits of e after its decimal point needed to contain all digit substrings of length n.
0

%I #46 Oct 04 2023 22:25:56

%S 20,371,8091,102127,1061612,12108840,198150340,1929504533

%N Number of decimal digits of e after its decimal point needed to contain all digit substrings of length n.

%C Length of the shortest prefix of the decimal expansion of e in which every possible digit string of length n occurs. We only consider the digits after the decimal point.

%C It is not known if every natural number appears in the decimal expansion of e. If this is the case, sequence a(n) contains a term for every n.

%F a(n) = A152182(n) + n - 2.

%e a(1) = 20, since 20 is the smallest number of digits in decimal expansion of e in with every digit 0..9 (or, to be more precise, every digit string of length 1) appears: 2.71828182845904523536.

%e a(2) = 371, since the first appearance of the digit string "12" is at positions 370-371 of the decimal expansion of e and the remaining digit strings of length 2 appear at least once before that position.

%e a(3) = 8091, since the first appearance of the digit string "548" is at positions 8089-8091 of the decimal expansion of e and the remaining digit strings of length 3 appear at least once before that position.

%e a(4) = 102127, since the first appearance of the digit string "1769" is at positions 102124-102127 of the decimal expansion of e and the remaining digit strings of length 4 appear at least once before that position.

%t dok = 300000; an = {};

%t For[li = 1, li <= 3, li++,

%t p = ToString[N[E, dok]];

%t cyf = {}; par = 0;

%t For[i = 3, i <= dok, i++,

%t If[par == 0,

%t a = StringTake[p, {i, i + li - 1}];

%t If[MemberQ[cyf, a] == False, cyf = Append[cyf, a];

%t If[Length[cyf] == 10^li, an = Append[an, i + li - 3]; par = 1]],

%t Break[]]

%t ]];

%t Print[an]

%Y Cf. A001113, A152182, A332262.

%K base,nonn,more

%O 1,1

%A _Bartlomiej Pawlik_, Sep 07 2023

%E a(6)-a(8) from _Michael S. Branicky_, Oct 04 2023