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, 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

Author

David James Sycamore, Jul 14 2023

Keywords

• nonn
• look

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A000005, A000027, A345147, A360179.

Programs

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 *)

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

Extensions

More terms from David A. Corneth, Jul 14 2023