OFFSET
1,3
COMMENTS
The numerators of the fraction F(n) = a(n)/A072044(n) may be generated directly by use of inclusion-exclusion; e.g., 1/4 + 1/9 + 1/25 - 1/225 - 1/100 - 1/36 + 1/900 = 9/25.
Following Euler, they may also be generated via products.
a(n)/A072044(n) -> 1 - 6/Pi^2 (provable via Euler, see references). This value is the supremal proportion of all rational number representations a/b that are reducible by some common factor (or, more broadly: the proportion of all pairs of integers a,b that are not coprime).
REFERENCES
Leonhard Euler, Introductio In Analysin Infinitorum Vol 1, 1748, p. 474.
LINKS
Leonhard Euler, Introductio in Analysin Infinitorum, Vol 1.
Wikipedia, Probability of coprimality.
FORMULA
a(n) = numerator(1 - Product_{k=1..n} (1 - 1/prime(k)^2)).
EXAMPLE
For n=3, 1 - (3/4)*(8/9)*(24/25) = 9/25.
Exactly 9/25 of all rational number representations are reducible by at least one prime factor of at most 5.
MAPLE
b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*(1-1/ithprime(n)^2)) end:
a:= n-> numer(1-b(n)):
seq(a(n), n=1..16); # Alois P. Heinz, May 11 2024
MATHEMATICA
a[n_]:=Numerator[1-Product[1-1/Prime[k]^2, {k, n}]]; Array[a, 16] (* Stefano Spezia, May 11 2024 *)
PROG
(PARI) a(n) = numerator(1 - prod(k=1, n, (prime(k)^2-1)/prime(k)^2)); \\ Michel Marcus, May 08 2024
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Brian Lee Burtner, May 08 2024
STATUS
approved