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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..58.

Matthew Russell Downey, On Bifurcations And Beauty (2023): pages 40-43.

Formula

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

Example

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.

Mathematica

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 *)

Prog

(Python)
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=", ")
(PARI) 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

Crossrefs

Indices of records in A018804. Cf. A000040.

Sequence in context: A070023 A035303 A291719 * A018609 A243390 A029461

Adjacent sequences: A360422 A360423 A360424 * A360426 A360427 A360428

Keyword

nonn

Author

Matthew Russell Downey, Feb 08 2023

Status

approved