A375581 Numbers m such that there exists an integer k >= 1 for which the concatenation of m, 2m, ..., km is an m-digit number.

1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 19, 20, 23, 27, 30, 33, 34, 37, 40, 43, 46, 49, 50, 53, 58, 59, 64, 69, 74, 79, 83, 84, 88, 93, 97, 103, 107, 111, 112, 116, 120, 124, 125, 129, 133, 137, 141, 146, 150, 154, 158, 162, 166, 167, 171, 175, 179, 183, 187, 191

Offset

1,2

Comments

Do there exist arbitrarily large gaps between successive terms?

Links

Table of n, a(n) for n=1..61.

Example

7 is a term because the concatenation of 7, 14, 21, 28 is 7142128 which has 7 digits.
21 is not a term because the concatenation of 21, 42, ..., 168 has 20 digits but concatenating this with 168+21 = 189 gives a number with 23 digits.

Mathematica

SelfIncrementingQ[n_] := Module[{len=Length@IntegerDigits[n], num, c=1, numDigits=0},
  numDigits = len*Ceiling[(10^len - n)/n];
  If[numDigits >= n, Return[Mod[n, len] == 0]];
  num = Ceiling[10^len/n]*n;
  While[numDigits < n + 1,
    If[(len + c)*Ceiling[(10^(len + c) - num)/n] >= n - numDigits,
      Return[Mod[n - numDigits, len + c] == 0],
      numDigits += (len + c)*Ceiling[(10^(len + c) - num)/n]
       ];
    num += Ceiling[(10^(len + c) - num)/n]*n;
    c++;
   ]
 ]
Select[Range[191], SelfIncrementingQ]

Crossrefs

Cf. A375461 (increment by 1).

Sequence in context: A077154 A077273 A032955 * A320848 A228234 A360792

Adjacent sequences: A375578 A375579 A375580 * A375582 A375583 A375584

Keyword

base,nonn

Author

Nicholas M. R. Frieler, Aug 19 2024

Status

approved