A366067 Trajectory of 578 under the map x -> A366144(x) (divide or multiply tau(x)).

578, 3468, 62424, 2996352, 359562240, 142386647040, 177698535505920, 45704355840, 61426654248960, 294847940395008000, 19688030208000, 71821934198784000, 5985161183232, 11491509471805440, 1773381091328, 978906362413056, 2443350280582987776, 265120473153536

Offset: 1

Author

Neal Gersh Tolunsky, Sep 28 2023

Keywords

• nonn

Comments

578 is the smallest starting number for this rule which seems to diverge (rather than entering a loop). This starting number was found by Robert Gerbicz, who made the following argument for the sequence's divergence:

When the exponent of 2 is t in the prime factorization of n, then t+1 divides tau(n), and if t+1 has a relatively large prime factor r, then it is likely that n is not divisible by r and so n will be multiplied by tau(n). So r is now a factor of n, which means it will be even harder to clear r from n. In the range of 2000 iterations, division occurs only a few times. In almost all cases, you need only check the exponent of 2 to see if it was a division or multiplication.

Examples

			a(1) = 578. Applying the rule in A366144, 578 has 6 divisors. 578 is not divisible by 6, so we multiply: a(2) = 578*6 = 3468.
a(7) = 177698535505920, which has 3888 divisors. 177698535505920 is divisible by 3888, so we divide: a(8) = 177698535505920/3888 = 45704355840.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..960
• Neal Gersh Tolunsky, Log plot of a(1..1000).

Crossrefs

Cf. A366144, A033950.

Programs

Mathematica

a[1] = 578; a[n_] := a[n] = a[n-1] * If[Divisible[a[n-1], d = DivisorSigma[0, a[n-1]]], 1/d, d]; Array[a, 18] (* Amiram Eldar, Sep 29 2023 *)

Python

from itertools import islice
from sympy import divisor_count
def f(n): return n//dn if n%(dn:=divisor_count(n)) == 0 else n*dn
def agen(x=578): # generator of terms
    while True: yield x; x = f(x)
print(list(islice(agen(), 18))) # Michael S. Branicky, Oct 03 2023

Python

from math import prod
from collections import Counter
from itertools import islice
from sympy import factorint
def A366067_gen(): # generator of terms
    a, b = 578, Counter({2:1,17:2})
    while True:
        yield a
        c = prod((e+1 for e in b.values()))
        if (d:=sum((Counter(factorint(e+1)) for e in b.values()),start=Counter()))<=b:
            a //= c
            b -=d
        else:
            a *= c
            b += d
A366067_list = list(islice(A366067_gen(),20)) # Chai Wah Wu, Oct 03 2023