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

Author

Nicholas M. R. Frieler, Aug 19 2024

Keywords

• base
• nonn

Comments

Do there exist arbitrarily large gaps between successive terms?

Examples

			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.
		

Crossrefs

Cf. A375461 (increment by 1).

Programs

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]