A362128 -id:A362128 - 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 *)

A362127 Records in A360179.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 19, 22, 28, 32, 34, 40, 43, 47, 51, 53, 56, 57, 60, 63, 64, 65, 69, 75, 77, 79, 80, 84, 86, 98, 99, 102, 105, 107, 119, 129, 130, 141, 143, 146, 150, 162, 170, 182, 188, 200, 202, 214, 220, 226, 232, 240, 248, 252, 262, 264, 268, 282, 294, 310, 326, 330, 342, 354, 362

Offset: 1

Michael De Vlieger, Apr 09 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..8238
Michael De Vlieger, Labeled scatterplot of m = A360179(n) for n = 1..512. Records, terms of this sequence, appear in red, local minima in blue, terms following novel m in gold and by extant m in green. The cyan line indicates the smallest number u not in a(1..n) and is not a feature of the sequence itself. The magenta line indicates the "prevailing low" L(n) and is not a feature of the sequence itself.

Cf. A000005, A360179, A362128.

nn = 10^4; c[] := False; h[] := 0; f[n_] := DivisorSigma[0, n];
a[1] = j = u = 1; r = 0;
Reap[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++];
    If[k > r, r = k; Sow[r]], {n, 2, nn}] ][[-1, -1]]

A362135 Indices of novel terms in A360179.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 15, 17, 21, 25, 29, 31, 36, 41, 47, 52, 58, 64, 69, 74, 82, 87, 92, 100, 106, 112, 120, 128, 138, 145, 153, 160, 168, 176, 183, 189, 198, 209, 220, 233, 242, 252, 265, 277, 285, 294, 309, 325, 338, 349, 359, 371, 384, 400, 413, 430, 447, 459, 474, 492, 515, 533, 545, 557, 570, 594

Offset: 1

Michael De Vlieger, Apr 10 2023

nonn

If we read A360179 as an irregular triangle of rows whose terms strictly increase, then a(n) is the index k such that A360179(k) is the last and largest term in row n.

A362128 is a subsequence of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..58188

Cf. A360179, A362128, A362134.

nn = 600;
c[] := False; h[] := 0; f[n_] := DivisorSigma[0, n];
a[1] = j = u = w = 1;
{1}~Join~Rest@ Reap[Do[
      If[c[j],
        k = j + f[u]; h[j]++; h[u]--,
        k = f[j]; c[j] = True; h[j]++; Sow[n - 1] ];
      u = Min[u, j]; Set[{a[n], q[k], j}, {k, True, k}];
      While[h[u] == 0, u++], {n, 2, nn}] ][[-1, -1]]

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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A362127 Records in A360179.

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A362135 Indices of novel terms in A360179.

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