cp's OEIS Frontend

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).

%I A309797 #33 Nov 14 2019 17:20:08
%S A309797 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,
%T A309797 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,
%U A309797 10,3,3,3,6,6,6,6,6,7,6,3,9,2,2,2,2,2,2
%N 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).
%C A309797 The sequence is well defined as we can always extend the sequence with a number that has not yet appeared.
%C A309797 The number 1 appears infinitely many times in the sequence:
%C A309797 - by contradiction: suppose that m is the index of the last occurrence of 1 in the sequence,
%C A309797 - there is no n > 0 such that n + k = m and n + 2*k = 2*m (with k > 0),
%C A309797 - so we can choose a(2*m) = 1, QED.
%C A309797 This sequence has connections with A003602:
%C A309797 - here we have up to a(n) numbers k such that a(n+k) = a(n+2*k), there we have no such numbers,
%C A309797 - 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),
%C A309797 - then f_v corresponds to A003602 where all but the first term have been repeated 2*v+1 times.
%H A309797 Rémy Sigrist, <a href="/A309797/b309797.txt">Table of n, a(n) for n = 1..10000</a>
%H A309797 Rémy Sigrist, <a href="/A309797/a309797.gp.txt">PARI program for A309797</a>
%H A309797 Rémy Sigrist, <a href="/A309797/a309797.png">Scatterplot of the first 500000 terms</a>
%F A309797 a(n) >= #{ k>0 such that a(n+k) = a(n+2*k) }.
%e A309797 The first terms, alongside the corresponding k's, are:
%e A309797   n   a(n)  k's
%e A309797   --  ----  ----------
%e A309797    1     1  {1}
%e A309797    2     1  {1}
%e A309797    3     1  {2}
%e A309797    4     1  {1}
%e A309797    5     2  {1, 3}
%e A309797    6     2  {2, 14}
%e A309797    7     2  {1, 2}
%e A309797    8     1  {1}
%e A309797    9     1  {1}
%e A309797   10     1  {4}
%e A309797   11     1  {1}
%e A309797   12     3  {2, 3, 68}
%e A309797   13     3  {1, 4, 22}
%e A309797   14     2  {1, 2}
%o A309797 (PARI) See Links section.
%Y A309797 Cf. A003602, A329268 (positions of 1).
%K A309797 nonn,look
%O A309797 1,5
%A A309797 _Rémy Sigrist_, Nov 11 2019