A249225 - OEIS

%I #36 Mar 17 2018 04:03:41

%S 1,2,9,8,125,18,343,352,1863,250,81191,1656,1118273,81634,271125,

%T 182144,41421503,400626,271211719,13579000,66392109,30408026,

%U 439143531,185794272,1998490625,1829999938,26311281597,10595920328,9146733078187

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

%C Let for instance a = 4 and b = 1, then the progression is p = [5, 9, 13, 17, 21, ...], and, with n = 1000 terms, one gets log(lcm(p)) = 2657.63... and A(a)*n = A(4)*1000 = (8/3)*1000 = 2666.66... (independent of b).

%H Vincenzo Librandi, <a href="/A249225/b249225.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="http://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, ...

%p f:= n -> numer(n/numtheory:-phi(n)*add(1/j, j = select(t -> igcd(t,n)=1, [$1..n-1]))):

%p f(1):= 1:

%p map(f, [$1..100]); # _Robert Israel_, Mar 14 2018

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

%o (PARI) a(n)={numerator(n*sum(j=1,n,if(gcd(j,n)==1, 1/j))/eulerphi(n))} \\ _Andrew Howroyd_, Mar 14 2018

%Y Cf. A249226 (denominators).

%K nonn,frac

%O 1,2

%A _Jean-François Alcover_, Oct 23 2014