A329735 a(n) is the least k > 0 such that the binary representation of n appears as a substring in the binary representation of at least half of the numbers in the range 1..k.

1, 2, 14, 38, 110, 62, 1006, 2206, 5072, 21504, 7114, 3704, 13868, 4058, 4067254, 4384886, 9535340, 39157714, 20466206, 5565048, 732167206, 47755164, 24722194, 12837030, 27081364, 14017192, 231845728, 15111866, 32273342, 16292028, 17478178355102

Offset

1,2

Comments

The sequence is well defined as for any n > 0, the proportion of numbers in the range 1..k whose binary representation contains that of n tends to 1 as k tends to infinity.

For any n > 0, the binary representation of n appears as a substring in the binary representation of a(n).

Apparently, records occur at indices n such that the representation of n in base 2^w contains only the digit 2^k for some w and k such that 0 <= k < w (see A330220).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..512

Rémy Sigrist, PARI program for A329735

Example

For n = 3:
- the binary representation of 3 is "11",
- the binary representation of the first numbers, alongside the proportion p of those containing "11", is:
  k   bin(k)  p
  --  ------  ----
   1       1     0
   2      10     0
   3      11   1/3
   4     100   1/4
   5     101   1/5
   6     110   1/3
   7     111   3/7
   8    1000   3/8
   9    1001   1/3
  10    1010  3/10
  11    1011  4/11
  12    1100  5/12
  13    1101  6/13
  14    1110   1/2
- we first reach a proportion p >= 1/2 for k = 14,
- hence a(3) = 14.

Prog

(PARI) See Links section.

Crossrefs

Cf. A330220.

Sequence in context: A092344 A281711 A216528 * A290124 A192349 A230801

Adjacent sequences: A329732 A329733 A329734 * A329736 A329737 A329738

Keyword

nonn,base

Author

Rémy Sigrist, Nov 20 2019

Status

approved