A365415 Number of decimal digits of e after its decimal point needed to contain all digit substrings of length n.

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

Offset: 1

Author

Bartlomiej Pawlik, Sep 07 2023

Keywords

• base
• nonn
• more

Comments

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.

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.

Examples

			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.
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.
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.
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.
		

Crossrefs

Cf. A001113, A152182, A332262.

Programs

Mathematica

dok = 300000; an = {};
For[li = 1, li <= 3, li++,
  p = ToString[N[E, dok]];
  cyf = {}; par = 0;
  For[i = 3, i <= dok, i++,
   If[par == 0,
    a = StringTake[p, {i, i + li - 1}];
    If[MemberQ[cyf, a] == False, cyf = Append[cyf, a];
     If[Length[cyf] == 10^li, an = Append[an, i + li - 3]; par = 1]],
    Break[]]
   ]];
Print[an]

Formula

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

Extensions

a(6)-a(8) from Michael S. Branicky, Oct 04 2023