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.
%I A236435 #26 Aug 26 2025 04:59:47
%S A236435 1,3,2,12,96,1152,2304,41472,165888,3981312,119439360,3822059520,
%T A236435 7644119040,321052999680,1284211998720,61642175938560,
%U A236435 3328677500682240,199720650040934400,399441300081868800,1597765200327475200,115039094423578214400,230078188847156428800,18406255107772514304000
%N A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).
%C A236435 A236436(n)/(a(n)*zeta(2)) is the asymptotic density of the prime(n-1)-rough squarefree numbers (squarefree numbers whose prime factors are all >= prime(n-1)) for n >= 2. E.g., A236436(2)/(a(2)*zeta(2)) = 2/(3*zeta(2)) = 4/Pi^2 (A185199) is the asymptotic density of the odd squarefree numbers (A056911), and A236436(3)/(a(3)*zeta(2)) = 1/(2*zeta(2)) = 3/Pi^2 (A104141) is the asymptotic density of the 5-rough squarefree numbers (A276378). - _Amiram Eldar_, Aug 26 2025
%D A236435 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979; Theorem 429.
%H A236435 Vincenzo Librandi, <a href="/A236435/b236435.txt">Table of n, a(n) for n = 1..200</a>
%H A236435 Jonathan Sondow and Eric W. Weisstein, <a href="https://mathworld.wolfram.com/MertensTheorem.html">MathWorld: Mertens Theorem</a>.
%F A236435 a(n+1) / A236436(n+1) = (A072045(n)/A072044(n)) / (A038110(n+1)/A060753(n+1)) because 1+x = (1-x^2) / (1-x).
%F A236435 a(n) / A236436(n) = Product_{k=1..n-1} (1 + 1/prime(k)) ~ (6/Pi^2)*exp(gamma)*log(n) as n -> infinity, by Mertens's theorem.
%e A236435 (1 + 1/2)*(1 + 1/3)*(1 + 1/5)*(1 + 1/7) = 96/35 has numerator a(5) = 96.
%e A236435 Fractions begin with 1, 3/2, 2, 12/5, 96/35, 1152/385, 2304/715, 41472/12155, 165888/46189, 3981312/1062347, 119439360/30808063, 3822059520/955049953, ...
%t A236435 Numerator@Table[ Product[ 1 + 1/Prime[ k], {k, 1, n-1}], {n, 1, 23}]
%Y A236435 Cf. A038110, A060753, A072044, A072045, A073004, A236436 (denominators), A335004.
%Y A236435 Cf. A013661, A056911, A104141, A185199, A276378.
%K A236435 nonn,frac,changed
%O A236435 1,2
%A A236435 _Jonathan Sondow_, Feb 01 2014