

A361511


a(1) = 1. Thereafter if a(n1) is a novel term, a(n) = d(a(n1)); otherwise, if a(n1) is the tth nonnovel term, a(n) = a(n1) + d(a(t)), where d is the divisor function A000005.


8



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, 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, 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
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OFFSET

1,3


COMMENTS

Inspired by A360179, but uses a simpler rule for nonnovel terms.
It is an obvious conjecture that every number eventually appears, but is there a proof?


LINKS

Michael De Vlieger, Scatterplot of a(n), 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.


EXAMPLE

The initial terms (in the third column, N = novel term, D = nonnovel term):
.n.a(n).....t
.1,..1,.N,
.2,..1,.D,..1
.3,..2,.N,
.4,..2,.D,..2
.5,..3,.N,
.6,..2,.D,..3
.7,..4,.N,
.8,..3,.D,..4
.9,..5,.N,
10,..2,.D,..5
11,..4,.D,..6
12,..6,.N,
13,..4,.D,..7
14,..7,.N,
15,..2,.D,..8
16,..4,.D,..9
17,..6,.D,.10
18,..8,.N,
19,..4,.D,.11
20,..7,.D,.12
21,.11,.N,
22,..2,.D,.13
...
If n=8, for example, a(8) = 3 is a nonnovel term, the 4th such, so a(9) = a(8) + d(a(4)) = 3 + d(2) = 5.
Can be read as an irregular triangle of increasing subsequences:
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;
2, 4, 6, 8, 10, 13, 15;
4, 8, 12;
6, 9, 13, 15, 17;
2, 4, 7, 11, 15, 19;
etc.
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MATHEMATICA

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 *)


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



