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
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Nov 20 2019
STATUS
approved