A249225 Numerators of constants A(a) related to the asymptotic LCM of arithmetic progressions a*n+b (a and b coprime).

1, 2, 9, 8, 125, 18, 343, 352, 1863, 250, 81191, 1656, 1118273, 81634, 271125, 182144, 41421503, 400626, 271211719, 13579000, 66392109, 30408026, 439143531, 185794272, 1998490625, 1829999938, 26311281597, 10595920328, 9146733078187

Offset

1,2

Comments

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).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Steven Finch, Cilleruelo's LCM Constants, 2013. [Cached copy, with permission of the author]

Eric Weisstein's MathWorld, Dirichlet's theorem

Wikipedia, Dirichlet's theorem on arithmetic progressions

Index entries for sequences related to lcm's

Formula

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

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

Example

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

Maple

f:= n -> numer(n/numtheory:-phi(n)*add(1/j, j = select(t -> igcd(t, n)=1, [$1..n-1]))):
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Mar 14 2018

Mathematica

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

Prog

(PARI) a(n)={numerator(n*sum(j=1, n, if(gcd(j, n)==1, 1/j))/eulerphi(n))} \\ Andrew Howroyd, Mar 14 2018

Crossrefs

Cf. A249226 (denominators).

Sequence in context: A272348 A351166 A092270 * A191351 A324553 A230283

Adjacent sequences: A249222 A249223 A249224 * A249226 A249227 A249228

Keyword

nonn,frac

Author

Jean-François Alcover, Oct 23 2014

Status

approved