A345147 -id:A345147 - cp's OEIS frontend

A360179 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise a(n) = a(n-1) + d(u), where d is the divisor function A000005 and u is the smallest unstarred prior term (each time we use a prior term we star it, and starred terms cannot be reused).

1, 1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 6, 4, 6, 8, 4, 7, 2, 5, 7, 10, 4, 7, 10, 12, 6, 8, 12, 16, 5, 9, 3, 5, 7, 9, 11, 2, 4, 6, 9, 13, 2, 4, 6, 9, 13, 15, 4, 8, 11, 15, 19, 2, 5, 7, 9, 11, 14, 4, 7, 10, 12, 15, 18, 6, 10, 14, 18, 22, 4, 8, 11

Offset: 1

David James Sycamore, Jan 29 2023

Whilst the definition is subtly different from that of A345147, d(u) being used in place of u, the scatterplots are remarkably different, the one for this sequence displaying numerous precipitous "gorges" which are open to explanation. 1 is the only number which occurs precisely twice, all other numbers are repeated infinitely many times.
From Michael De Vlieger, Apr 04 2023: (Start)
The sequence is a series of nondecreasing cycles that reach a maximum M and then reset to start a new cycle. (See scatterplot B.)
The sequence is dynamic and responds to a bank of copies of the same number called a "prevailing low" L. When M < L, the sequence experiences a run of short or "crashed" cycles that make no headway at eliminating the copies of L, resulting in a "gorge" in the scatterplot.
Referring to scatterplot A:
The green line represents the smallest missing number u and is not actually a feature of the sequence. The red line represents the "prevailing low" L(n), also is not a feature of the sequence.
Dark blue terms a(n) = tau(a(n-1))..421 populate a "semi-coherent" phase (1A) of cycle c(i), where tau(n) = A000005(n).
Light blue terms a(n) = 422..L populate the "coherent" phase (1B) of cycle c(i). Black terms m > L populate phase (2) of c(i).
Magenta terms constitute a crashed cycle that has M < L; multiple consecutive crashed cycles constitute a gorge. In crashed cycles, we have only phase (1).
The "triple point" of the graph, where we first have phase (1B), appears to be a(14478) = 414, but is in actuality (given 2^20 terms) a(14786) = 422. (End)

			a(2) = 1 since a(1) = 1 is a novel term and d(1) = 1. Thus the sequence starts 1,1 and since a(2) is a repeated term, a(3) = a(2) + d(1) (1 = least unstarred prior term). Therefore a(3) = 1 + 1 = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..40000 [The exceptionally large b-file was added at my request. - _N. J. A. Sloane_, Apr 07 2023]
Michael De Vlieger, Scatterplot A of a(n), n = 1..40000.
Michael De Vlieger, Scatterplot B of a(n) for n = 1..128. Records appear in red, local minima in blue, terms instigated by a(n) = m new to the sequence appear in gold, otherwise in green. The magenta line indicates the smallest missing number u not in a(1..n-1).
Michael De Vlieger, Scatterplot of a(n), n = 1500000.

Cf. A000005, A343887, A345147.
Cf. A362127 records, A362128 indices of records.
Cf. A362129 a(n) mod 2, A362130 d(a(n)) mod 2.
Cf. A362131 smallest missing number in a(1..n).
Cf. A362134 novel terms, A362135 indices of novel terms.
Cf. A362136 row lengths, if this sequence seen as rows of strictly increasing terms.
See A361511 for another version.

nn = 120; c[] := False; h[] := 0; f[n_] := DivisorSigma[0, n]; a[1] = j = u = 1; Do[If[c[j], k = j + f[u]; h[j]++; h[u]--, k = f[j]; c[j] = True; h[j]++]; u = Min[u, j]; Set[{a[n], j}, {k, k}]; While[h[u] == 0, u++], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Feb 02 2023 *)

A364236 a(1) = 1. For n > 1, if a(n-1) is a novel term, a(n) = d(a(n-1)), else if a(n-1) is a repeat term seen k (>1) times, a(n) = a(n-1) + d(k-1), where d is the divisor counting function A000005.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 5, 2, 4, 6, 4, 6, 7, 2, 5, 6, 8, 4, 7, 8, 9, 3, 5, 7, 9, 10, 4, 6, 8, 10, 11, 2, 4, 8, 10, 12, 6, 9, 11, 12, 13, 2, 6, 8, 11, 13, 14, 4, 6, 10, 12, 14, 15, 4, 8, 10, 13, 15, 16, 5, 7, 9, 11, 13, 15, 17, 2, 4, 7, 10, 12, 14, 16, 17, 18

Offset: 1

David James Sycamore, Jul 14 2023

In other words the appearance of a novel term m introduces d(m) as next term, whereas the appearance of repeat term m introduces m + the number of divisors of the number of repetitions of m.
1 is seen only twice, but all other numbers appear infinitely many times.
Prime terms may appear in 3 different ways: consequent to the second appearance of p-1, to the first appearance of m where d(m) = p, or to a repeat term m seen k (>1) times, where m + d(k-1) = p.
The plot consists of consecutive strictly increasing trajectories starting d(r(k)) after the k_th record term r(k), and ending with r(k+1) = r(k)+1, meaning that records are given by A000027. This behavior, which determines the smooth whaleback shape of the plot is open to explanation.

			a(1) = 1 is a novel term so a(2) = d(a(1)) = d(1) = 1.
Since 1 has been repeated once, a(3) = 1 + d(1) = 2.
a(3) = 2 introduces a(4) = d(2) = 2 and so on.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot of a(n), n = 1..256, showing chains c(k) beginning with tau(k-1) and strictly increasing until we reach k itself. We highlight chain minimum tau(k-1) in blue and maximum k in red.
Michael De Vlieger, Plot of a(n), n = 1..2^16, showing fine structure.
Michael De Vlieger, Plot of a(n) n = 1..2^20, showing aggregate structure.

Cf. A000005, A000027, A345147, A360179.

nn = 120; c[] := 0; a[1] = j = 1; f[x] := DivisorSigma[0, x]; Do[k = If[# == 0, f[j], j + f[#] ] &[c[j]]; c[j]++; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Jul 14 2023 *)

lista(nn) = {my(va=vector(nn)); va[1] = 1; for (n=2, nn, my(vv = Vec(va, n-1)); my(k = #select(x->(x==va[n-1]), vv)); if (k==1, va[n] = numdiv(va[n-1]), va[n] = va[n-1] + numdiv(k-1));); va;} \\ Michel Marcus, Jul 14 2023

More terms from David A. Corneth, Jul 14 2023

cp's OEIS Frontend

A360179 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise a(n) = a(n-1) + d(u), where d is the divisor function A000005 and u is the smallest unstarred prior term (each time we use a prior term we star it, and starred terms cannot be reused).

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