A338763 - OEIS

%I #14 Nov 09 2020 00:29:22

%S 3,6,9,13,15,17,21,25,28,29,33,37,38,41,45,49,53,54,57,61,65,66,69,70,

%T 73,77,81,85,86,89,93,97,101,102,105,109,113,117,118,120,121,125,129,

%U 133,134,137,141,145,149,150,153,156,157,161,165,166,169,173,177

%N Let L_1 = (1, 2, ...); for any n > 0, let M_n = Min_{k > 0} L_n(k) + L_n(k+1) and K_n = Min_{ k | L_n(k) + L_n(k+1) = M_n }, L_{n+1} is obtained by replacing the two terms L_n(K_n) and L_n(K_n+1) by M_n in L_n; a(n) = M_n.

%C In other words, we start with the natural numbers and repeatedly replace the leftmost pair of consecutive terms with minimal sum by its sum; a(n) corresponds to the sum at n-th step.

%C This sequence is weakly increasing, and tends to infinity (as for any m > 0, there are only finitely many runs of two or more consecutive integers with a sum < m).

%H Rémy Sigrist, <a href="/A338763/b338763.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A338763/a338763.gp.txt">PARI program for A338763</a>

%F a(n) <= a(n+1).

%e The first terms, alongside L_n, are:

%e   n   a(n)  L_n

%e   --  ----  ----------------------------------------------------------

%e    1     3  { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... }

%e    2     6  {   3,  3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... }

%e    3     9  {      6,  4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... }

%e    4    13  {      6,    9,  6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... }

%e    5    15  {      6,    9,   13,  8, 9, 10, 11, 12, 13, 14, 15, ... }

%e    6    17  {         15,     13,  8, 9, 10, 11, 12, 13, 14, 15, ... }

%e    7    21  {         15,     13,   17,  10, 11, 12, 13, 14, 15, ... }

%e    8    25  {         15,     13,   17,    21,   12, 13, 14, 15, ... }

%e    9    28  {         15,     13,   17,    21,     25,   14, 15, ... }

%e   10    29  {             28,       17,    21,     25,   14, 15, ... }

%o (PARI) See Links section.

%Y Cf. A326936, A338764.

%K nonn

%O 1,1

%A _Rémy Sigrist_, Nov 07 2020