A354606 a(1) = 1; for n > 1, a(n) is number of terms in the first n-1 terms of the sequence that have the same number of divisors as a(n-1).

1, 1, 2, 1, 3, 2, 3, 4, 1, 4, 2, 5, 6, 1, 5, 7, 8, 2, 9, 3, 10, 3, 11, 12, 1, 6, 4, 4, 5, 13, 14, 5, 15, 6, 7, 16, 1, 7, 17, 18, 2, 19, 20, 3, 21, 8, 9, 6, 10, 11, 22, 12, 4, 7, 23, 24, 1, 8, 13, 25, 8, 14, 15, 16, 2, 26, 17, 27, 18, 5, 28, 6, 19, 29, 30, 2, 31, 32, 7, 33, 20, 8, 21, 22, 23, 34

Offset: 1

Author

Scott R. Shannon, Jul 08 2022

Keywords

• nonn
• easy

Comments

After 250000 terms the most common number of divisors of all terms are 4, 8, 2, 12, 16 divisors. These correspond to the uppermost five lines of the attached image. It is unknown if these stay the most common or are passed by numbers with more divisors as n gets arbitrarily large.

See A355606 for the indices where a(n) = 1.

Examples

			a(6) = 2 as a(5) = 3 which has two divisors, and the total number of terms in the first five terms with two divisors is two, namely a(3) = 2 and a(5) = 3.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Crossrefs

Cf. A355606, A000005, A124056, A342585.

Programs

Mathematica

nn = 120; c[] := 0; a[1] = j = 1; Do[k = ++c[DivisorSigma[0, j]]; Set[{a[n], j}, {k, k}], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Dec 12 2024 *)

Python

from sympy import divisor_count
from collections import Counter
def f(n): return divisor_count(n)
def aupton(nn):
    an, fan, alst, inventory = 1, 1, [1], Counter([1])
    for n in range(2, nn+1):
        an = inventory[fan]
        fan = f(an)
        alst.append(an)
        inventory.update([fan])
    return alst
print(aupton(86)) # Michael S. Branicky, Jul 09 2022