A164935 a(n) is the smallest number x such that the decimal representation of n appears as a substring of the decimal representations of the numbers [1...x] >= x times.

100559404366, 1, 28263827, 371599983, 499999984, 5555555555, 6666666666, 7777777777, 8888888888, 9999999999, 109999999999999999999999999999999999999999999999999999999999999999999999999999999999999999810

Offset

0,1

Comments

Starting from n = 2, a(n) = min(A163500,A164321).

Links

Table of n, a(n) for n=0..10.

IBM Corp., April 2004 "Ponder This" challenge.

Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 6.

Yet another Google puzzle

Mathematica

cz[n_, k_] := Floor[n/10^k] 10^(k - 1) + (Ceiling[Floor[n/10^(k - 1)]/10] - Floor[Floor[n/10^(k - 1)]/10] - 1) (10^(k - 1) - Mod[n, 10^(k - 1)] - 1) countZeroes[n_] := (z = 0; k = 1; len = Length[IntegerDigits[n]]; While[k < len, z = z + cz[n, k]; k++ ]; z) c = 8; d = 16; While[d - c > 1 , If[countZeroes[d] >= c, d = (c + d)/2, {c, d} = {d, d + 2 d - 2 c}]]; While[ countZeroes[c] < c, c++ ]; Print[c] countAny[n_, anyK_] := (z = 0; lenK = Length[IntegerDigits[anyK]]; len = Length[IntegerDigits[n]]; k = lenK;
While[k <= len, middle = Mod[Floor[n/10^(k - lenK)], 10^lenK]; If [middle > anyK, z = z + ( Floor[n/10^k] + 1) 10^(k - lenK)]; If[middle < anyK, z = z + Floor[n/10^k] 10^(k - lenK)]; If[middle == anyK, z = z + Floor[n/10^k] 10^(k - lenK) + Mod[n, 10^(k - lenK)] + 1]; k++ ]; z) i = 1; c = 8; d = 16; While[i < 20, While[d - c > 1 , If[countAny[d, i] >= c, d = (c + d)/2, {c, d} = {d, d + 2 d - 2 c}]]; While[countAny[c, i] < c, c++ ]; Print[c]; d = c + 8; i++ ]

Crossrefs

Cf. A163500, A164321, A092175.

Sequence in context: A072145 A192108 A214702 * A349321 A168340 A104799

Adjacent sequences: A164932 A164933 A164934 * A164936 A164937 A164938

Keyword

base,nonn,uned

Author

Tanya Khovanova and Gregory Marton, Aug 31 2009

Status

approved