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A361511 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. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Inspired by A360179, but uses a simpler rule for non-novel 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 = non-novel 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 non-novel term, the 4th such, so a(9) = a(8) + d(a(4)) = 3 + d(2) = 5.
Comment from Michael De Vlieger, Apr 08 2023 (Start)
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.
(End)
The rows end with the novel terms - see A361512, A361513 - and their lengths are given by A361514.
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
Sequence in context: A246029 A245564 A360179 * A345147 A214126 A205378
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 08 2023
STATUS
approved

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Last modified May 29 03:48 EDT 2024. Contains 372921 sequences. (Running on oeis4.)