A249226 - OEIS

%I #18 Feb 16 2025 08:33:24

%S 1,1,4,3,48,5,120,105,560,63,25200,385,332640,19305,64064,45045,

%T 11531520,85085,73513440,2909907,14780480,6613425,113809696,37182145,

%U 475931456,386122275,5949143200,2151252675,2248776129600,215656441

%N Denominators of constants A(a) related to the asymptotic LCM of arithmetic progressions a*n+b (a and b coprime).

%H Andrew Howroyd, <a href="/A249226/b249226.txt">Table of n, a(n) for n = 1..1000</a>

%H Steven Finch, <a href="/A003418/a003418.pdf">Cilleruelo's LCM Constants</a>, 2013. [Cached copy, with permission of the author]

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DirichletsTheorem.html">Dirichlet's theorem</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions">Dirichlet's theorem on arithmetic progressions</a>

%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>

%F A(a) = (a/phi(a))*Sum_{j=1..a, gcd(j,a)=1} 1/j.

%F log(lcm_{k=0..n} a*k+b) ~ A(a)*n for gcd(a,b)=1.

%e Sequence A(a) begins 1, 2, 9/4, 8/3, 125/48, 18/5, 343/120, 352/105, 1863/560, ...

%t A[a_] := (a/EulerPhi[a])*Sum[If[GCD[j, a] == 1, 1/j, 0], {j, 1, a}]; Array[A, 40] // Denominator

%o (PARI) a(n)={denominator(n*sum(j=1, n, if(gcd(j, n)==1, 1/j))/eulerphi(n))} \\ _Andrew Howroyd_, Mar 16 2018

%Y Cf. A249225 (numerators).

%K nonn,frac

%O 1,3

%A _Jean-François Alcover_, Oct 23 2014