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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Brian Lee Burtner

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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).

Original entry on oeis.org

1, 1, 9, 457, 11213, 273347, 79439651, 5761023199, 277886746829, 33449007905699, 32197332748181219, 2322572170370125769, 3907895853135787075289, 657439531892484346088851, 63187594618979703535273733, 13660992716321028635960170769
Offset: 1

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Author

Brian Lee Burtner, May 08 2024

Keywords

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.

Examples

			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.

References

  • Leonhard Euler, Introductio In Analysin Infinitorum Vol 1, 1748, p. 474.

Links

Crossrefs

Cf. A072044, A072045, A229099.

Programs

  • 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 *)
  • PARI
    a(n) = numerator(1 - prod(k=1, n, (prime(k)^2-1)/prime(k)^2)); \\ Michel Marcus, May 08 2024

Formula

a(n) = numerator(1 - Product_{k=1..n} (1 - 1/prime(k)^2)).

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