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User: Matthew Russell Downey

Matthew Russell Downey has authored 2 sequences.

A360425 Indices of records in A018804.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 105, 108, 120, 140, 144, 168, 180, 210, 240, 252, 270, 280, 288, 300, 330, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 990, 1008, 1080, 1200, 1260, 1440

Offset: 1

Matthew Russell Downey, Feb 08 2023

nonn

a(n) seems to be divisible by any positive integer as n approaches infinity. Example: There seem to be 6 terms without a 2 (i.e., 1, 3, 5, 9, 15, 105), 14 terms without a 3 (as large as 280), 22 terms without a factor of 4 (as large as 6930), and 29 terms without a 5 (as large as 3276). True for a(n) < 10^8.
For any term a(n) > 1, it seems that there exists at least one term a(x) such that a(x) * prime(y) = a(n) where prime(y) <= (7/2) * A053669(a(x)). True for a(n) < 10^8.
The ratio of prime(n)/a(n) for indices {1..8, 11, 15, 17} is almost 2 (with an error of at most 1 on the numerator). The exact ratios are 2/1, 3/2, 5/3, 7/4, 11/5, 13/6, 17/8, 19/9, 31/15, 47/24, 59/30. - Matthew Russell Downey, Jul 25 2023

			A018804(36) = 168 is the largest value among the first 36 terms of A018804, so 36 is a term here; since it is the 18th value that sets a new record, a(18) = 36.

David A. Corneth, Table of n, a(n) for n = 1..286
Matthew Russell Downey, On Bifurcations And Beauty (2023): pages 40-43.

Indices of records in A018804. Cf. A000040.

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ (f @@@ FactorInteger[n]); seq[nmax_] := Module[{p, pm = 0, s = {}}, Do[If[(p = pil[n]) > pm, pm = p; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[1200] (* Amiram Eldar, Feb 13 2023 *)

f(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ A018804
lista(nn) = my(r=0, list=List()); for (n=1, nn, my(m=f(n)); if (m > r, listput(list, n); r = m);); Vec(list); \\ Michel Marcus, Feb 26 2023

from sympy import factorint
from math import prod
def A018804(m): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(m).items())
record = 0
for m in range(1, 2000):
  value = A018804(m)
  if value > record:
    record = value
    print(m, end=", ")

Conjecture: a(n) ~ exp(4*(n-1)/21). - Matthew Russell Downey, Jul 25 2023

A339614 Inputs n that yield a record-breaking value of A008908(n)/(log_2(n)+1) for the Collatz conjecture.

Original entry on oeis.org

1, 3, 7, 9, 27, 26623, 35655, 52527, 142587, 156159, 230631, 626331, 837799, 1723519, 3542887, 3732423, 5649499, 6649279, 8400511, 63728127, 3743559068799, 100759293214567, 104899295810901231

Offset: 1

Matthew Russell Downey, Dec 10 2020

The metric A008908(n)/(log_2(n)+1) is always equal to 1 for any power of 2 (where 1 is the smallest possible value).

			a(1) = 1, which is trivial, because the first element in any sequence is record setting.
a(5) = 27, because A008908(n)/(log_2(n)+1) yields a maximum value at n=27 among the first 27 elements, and there are 4 record-breaking elements beforehand.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A008908, A006877, A033492.

import math
oeis_A006877 = [1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 10623, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, 1117065, 1501353, 1723519, 2298025, 3064033, 3542887, 3732423, 5649499, 6649279, 8400511, 11200681, 14934241, 15733191, 31466382, 36791535, 63728127]
def stopping_time(n):
    time = 1
    while n>1:
        n = 3*n + 1 if n & 1 else n//2
        time += 1
    return time
def stopping_time_metric(n):
    time = stopping_time(n)
    logarithmic_distance = (math.log(n, 2)+1)
    return float(time/logarithmic_distance)
result = []
record_input = oeis_A006877[0]
record_stopping_time_metric = stopping_time_metric(record_input)
result.append(record_input)
for n in range(1, len(oeis_A006877)):
    current_input = oeis_A006877[n]
    current_stopping_time_metric = stopping_time_metric(current_input)
    if current_stopping_time_metric > record_stopping_time_metric:
        record_input = current_input
        record_stopping_time_metric = current_stopping_time_metric
        result.append(record_input)
for n in range(len(result)):
    print(result[n], end=", ")

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User: Matthew Russell Downey

A360425 Indices of records in A018804.

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