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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.

%I A329735 #15 Dec 08 2019 09:53:12
%S A329735 1,2,14,38,110,62,1006,2206,5072,21504,7114,3704,13868,4058,4067254,
%T A329735 4384886,9535340,39157714,20466206,5565048,732167206,47755164,
%U A329735 24722194,12837030,27081364,14017192,231845728,15111866,32273342,16292028,17478178355102
%N 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.
%C A329735 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.
%C A329735 For any n > 0, the binary representation of n appears as a substring in the binary representation of a(n).
%C A329735 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).
%H A329735 Rémy Sigrist, <a href="/A329735/b329735.txt">Table of n, a(n) for n = 1..512</a>
%H A329735 Rémy Sigrist, <a href="/A329735/a329735.gp.txt">PARI program for A329735</a>
%e A329735 For n = 3:
%e A329735 - the binary representation of 3 is "11",
%e A329735 - the binary representation of the first numbers, alongside the proportion p of those containing "11", is:
%e A329735   k   bin(k)  p
%e A329735   --  ------  ----
%e A329735    1       1     0
%e A329735    2      10     0
%e A329735    3      11   1/3
%e A329735    4     100   1/4
%e A329735    5     101   1/5
%e A329735    6     110   1/3
%e A329735    7     111   3/7
%e A329735    8    1000   3/8
%e A329735    9    1001   1/3
%e A329735   10    1010  3/10
%e A329735   11    1011  4/11
%e A329735   12    1100  5/12
%e A329735   13    1101  6/13
%e A329735   14    1110   1/2
%e A329735 - we first reach a proportion p >= 1/2 for k = 14,
%e A329735 - hence a(3) = 14.
%o A329735 (PARI) See Links section.
%Y A329735 Cf. A330220.
%K A329735 nonn,base
%O A329735 1,2
%A A329735 _Rémy Sigrist_, Nov 20 2019