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A372634
Numerators of the reduced fraction of all rational number representations (a/b, with a and b being integers) which can themselves be reduced by at least one common prime factor of at most prime(n).
0
1, 1, 9, 457, 11213, 273347, 79439651, 5761023199, 277886746829, 33449007905699, 32197332748181219, 2322572170370125769, 3907895853135787075289, 657439531892484346088851, 63187594618979703535273733, 13660992716321028635960170769
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).
A072045(n)/A072044(n) gives the complementary proportion of all rational number representations that are irreducible by any prime factor of at most A000040(n). This analogously converges to 6/Pi^2, the infimal proportion of all rational number representations a/b that are simply irreducible.
REFERENCES
Leonhard Euler, Introductio In Analysin Infinitorum Vol 1, 1748, p. 474.
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