

A309797


Lexicographically earliest sequence of positive integers such that for any n > 0 there are no more than a(n) numbers k > 0 such that a(n + k) = a(n + 2*k).


4



1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 2, 4, 2, 4, 5, 4, 1, 1, 1, 1, 3, 3, 3, 1, 6, 6, 6, 7, 5, 3, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 7, 5, 7, 5, 8, 5, 8, 8, 8, 8, 3, 1, 1, 1, 1, 9, 9, 6, 1, 1, 1, 1, 10, 3, 3, 3, 6, 6, 6, 6, 6, 7, 6, 3, 9, 2, 2, 2, 2, 2, 2
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OFFSET

1,5


COMMENTS

The sequence is well defined as we can always extend the sequence with a number that has not yet appeared.
The number 1 appears infinitely many times in the sequence:
 by contradiction: suppose that m is the index of the last occurrence of 1 in the sequence,
 there is no n > 0 such that n + k = m and n + 2*k = 2*m (with k > 0),
 so we can choose a(2*m) = 1, QED.
This sequence has connections with A003602:
 here we have up to a(n) numbers k such that a(n+k) = a(n+2*k), there we have no such numbers,
 for any v >= 0, let f_v be the lexicographically earliest sequence of positive integers such that there are no more than v numbers k such that f_v(n + k) = f_v(n + 2*k),
 then f_v corresponds to A003602 where all but the first term have been repeated 2*v+1 times.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A309797
Rémy Sigrist, Scatterplot of the first 500000 terms


FORMULA

a(n) >= #{ k>0 such that a(n+k) = a(n+2*k) }.


EXAMPLE

The first terms, alongside the corresponding k's, are:
n a(n) k's
  
1 1 {1}
2 1 {1}
3 1 {2}
4 1 {1}
5 2 {1, 3}
6 2 {2, 14}
7 2 {1, 2}
8 1 {1}
9 1 {1}
10 1 {4}
11 1 {1}
12 3 {2, 3, 68}
13 3 {1, 4, 22}
14 2 {1, 2}


PROG

(PARI) See Links section.


CROSSREFS

Cf. A003602, A329268 (positions of 1).
Sequence in context: A029432 A073426 A232439 * A199840 A126067 A327490
Adjacent sequences: A309794 A309795 A309796 * A309798 A309799 A309800


KEYWORD

nonn,look


AUTHOR

Rémy Sigrist, Nov 11 2019


STATUS

approved



