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

Author

Matthew Russell Downey, Feb 08 2023

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Indices of records in A018804. Cf. A000040.

Programs

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 *)

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

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=", ")

Formula

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