A361511 - OEIS

%I #38 Apr 09 2023 11:29:41

%S 1,1,2,2,3,2,4,3,5,2,4,6,4,7,2,4,6,8,4,7,11,2,5,7,9,3,6,10,4,8,11,13,

%T 2,4,6,8,10,13,15,4,8,12,6,9,13,15,17,2,4,7,11,15,19,2,4,8,11,15,21,4,

%U 8,11,13,17,19,21,24,8,10,12,16,5,7,9,12,16,18,6,10,14,4,7,11,13,15,17,19,23

%N a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise, if a(n-1) is the t-th non-novel term, a(n) = a(n-1) + d(a(t)), where d is the divisor function A000005.

%C Inspired by A360179, but uses a simpler rule for non-novel terms.

%C It is an obvious conjecture that every number eventually appears, but is there a proof?

%H Michael De Vlieger, <a href="/A361511/b361511.txt">Table of n, a(n) for n = 1..40000</a>

%H Michael De Vlieger, <a href="/A361511/a361511.png">Scatterplot of a(n)</a>, n = 1..2^16, showing records in red, smallest missing numbers in blue (small until they enter sequence, then large), terms deriving from novel predecessors in gold, otherwise green.

%H Michael De Vlieger, <a href="/A361511/a361511_1.png">Plot that shows the increasing subsequences that form the rows when the sequence is regarded as an irregular triangle</a>

%e The initial terms (in the third column, N = novel term, D = non-novel term):

%e   .n.a(n).....t

%e   .1,..1,.N,

%e   .2,..1,.D,..1

%e   .3,..2,.N,

%e   .4,..2,.D,..2

%e   .5,..3,.N,

%e   .6,..2,.D,..3

%e   .7,..4,.N,

%e   .8,..3,.D,..4

%e   .9,..5,.N,

%e   10,..2,.D,..5

%e   11,..4,.D,..6

%e   12,..6,.N,

%e   13,..4,.D,..7

%e   14,..7,.N,

%e   15,..2,.D,..8

%e   16,..4,.D,..9

%e   17,..6,.D,.10

%e   18,..8,.N,

%e   19,..4,.D,.11

%e   20,..7,.D,.12

%e   21,.11,.N,

%e   22,..2,.D,.13

%e ...

%e If n=8, for example, a(8) = 3 is a non-novel term, the 4th such, so a(9) = a(8) + d(a(4)) = 3 + d(2) = 5.

%e Comment from _Michael De Vlieger_, Apr 08 2023 (Start)

%e Can be read as an irregular triangle of increasing subsequences:

%e   1;

%e   1, 2;

%e   2, 3;

%e   2, 4;

%e   3, 5;

%e   2, 4,  6;

%e   4, 7;

%e   2, 4,  6, 8;

%e   4, 7, 11;

%e   2, 5,  7, 9;

%e   3, 6, 10;

%e   4, 8, 11, 13;

%e   2, 4,  6,  8, 10, 13, 15;

%e   4, 8, 12;

%e   6, 9, 13, 15, 17;

%e   2, 4,  7, 11, 15, 19;

%e   etc.

%e (End)

%e The rows end with the novel terms - see A361512, A361513 - and their lengths are given by A361514.

%t nn = 120; c[_] = False; f[n_] := DivisorSigma[0, n]; a[1] = m = 1; Do[(If[c[#], a[n] = # + f[a[m]] ; m++, a[n] = f[#] ]; c[#] = True) &[a[n - 1]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Apr 08 2023 *)

%Y Cf. A000005, A360179, A361512-A361516, A362095.

%K nonn,tabf

%O 1,3

%A _N. J. A. Sloane_, Apr 08 2023