A330220 -id:A330220 - cp's OEIS frontend

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

Rémy Sigrist, Nov 20 2019

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).

			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.

Rémy Sigrist, Table of n, a(n) for n = 1..512
Rémy Sigrist, PARI program for A329735

Cf. A330220.

PARI
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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.

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