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

1,1

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.

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..960

Neal Gersh Tolunsky, Log plot of a(1..1000).

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A366144, A033950.

Sequence in context: A252384 A252385 A158369 * A232888 A035754 A107550

Adjacent sequences: A366064 A366065 A366066 * A366068 A366069 A366070

Keyword

nonn

Author

Neal Gersh Tolunsky, Sep 28 2023

Status

approved