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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

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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.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

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.

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 *)

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])