cp's OEIS Frontend

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

%I A375581 #9 Sep 13 2024 07:37:18
%S A375581 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,
%T A375581 58,59,64,69,74,79,83,84,88,93,97,103,107,111,112,116,120,124,125,129,
%U A375581 133,137,141,146,150,154,158,162,166,167,171,175,179,183,187,191
%N A375581 Numbers m such that there exists an integer k >= 1 for which the concatenation of m, 2m, ..., km is an m-digit number.
%C A375581 Do there exist arbitrarily large gaps between successive terms?
%e A375581 7 is a term because the concatenation of 7, 14, 21, 28 is 7142128 which has 7 digits.
%e A375581 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.
%t A375581 SelfIncrementingQ[n_] := Module[{len=Length@IntegerDigits[n],num,c=1,numDigits=0},
%t A375581   numDigits = len*Ceiling[(10^len - n)/n];
%t A375581   If[numDigits >= n, Return[Mod[n, len] == 0]];
%t A375581   num = Ceiling[10^len/n]*n;
%t A375581   While[numDigits < n + 1,
%t A375581     If[(len + c)*Ceiling[(10^(len + c) - num)/n] >= n - numDigits,
%t A375581       Return[Mod[n - numDigits, len + c] == 0],
%t A375581       numDigits += (len + c)*Ceiling[(10^(len + c) - num)/n]
%t A375581        ];
%t A375581     num += Ceiling[(10^(len + c) - num)/n]*n;
%t A375581     c++;
%t A375581    ]
%t A375581  ]
%t A375581 Select[Range[191],SelfIncrementingQ]
%Y A375581 Cf. A375461 (increment by 1).
%K A375581 base,nonn
%O A375581 1,2
%A A375581 _Nicholas M. R. Frieler_, Aug 19 2024