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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.
1
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
1,3
COMMENTS
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.
LINKS
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.
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
KEYWORD
nonn,look
AUTHOR
EXTENSIONS
More terms from David A. Corneth, Jul 14 2023
STATUS
approved