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

1,3

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.

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.

Example

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.

Prog

(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

Crossrefs

Cf. A355606, A000005, A124056, A342585.

Sequence in context: A322391 A201592 A363757 * A362031 A234361 A350013

Adjacent sequences: A354603 A354604 A354605 * A354607 A354608 A354609

Keyword

nonn

Author

Scott R. Shannon, Jul 08 2022

Status

approved