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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).

%I A372634 #33 Jun 01 2024 18:22:55
%S A372634 1,1,9,457,11213,273347,79439651,5761023199,277886746829,
%T A372634 33449007905699,32197332748181219,2322572170370125769,
%U A372634 3907895853135787075289,657439531892484346088851,63187594618979703535273733,13660992716321028635960170769
%N 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).
%C A372634 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.
%C A372634 Following Euler, they may also be generated via products.
%C A372634 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).
%C A372634 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.
%D A372634 Leonhard Euler, Introductio In Analysin Infinitorum Vol 1, 1748, p. 474.
%H A372634 Leonhard Euler, <a href="https://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch15vol1.pdf">Introductio in Analysin Infinitorum, Vol 1</a>.
%H A372634 Wikipedia, <a href="https://en.wikipedia.org/wiki/Coprime_integers#Probability_of_coprimality">Probability of coprimality</a>.
%F A372634 a(n) = numerator(1 - Product_{k=1..n} (1 - 1/prime(k)^2)).
%e A372634 For n=3, 1 - (3/4)*(8/9)*(24/25) = 9/25.
%e A372634 Exactly 9/25 of all rational number representations are reducible by at least one prime factor of at most 5.
%p A372634 b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*(1-1/ithprime(n)^2)) end:
%p A372634 a:= n-> numer(1-b(n)):
%p A372634 seq(a(n), n=1..16);  # _Alois P. Heinz_, May 11 2024
%t A372634 a[n_]:=Numerator[1-Product[1-1/Prime[k]^2,{k,n}]]; Array[a,16] (* _Stefano Spezia_, May 11 2024 *)
%o A372634 (PARI) a(n) = numerator(1 - prod(k=1, n, (prime(k)^2-1)/prime(k)^2)); \\ _Michel Marcus_, May 08 2024
%Y A372634 Cf. A072044, A072045, A229099.
%K A372634 nonn,frac
%O A372634 1,3
%A A372634 _Brian Lee Burtner_, May 08 2024