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A249225
Numerators of constants A(a) related to the asymptotic LCM of arithmetic progressions a*n+b (a and b coprime).
2
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
Steven Finch, Cilleruelo's LCM Constants, 2013. [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Dirichlet's theorem
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
KEYWORD
nonn,frac
AUTHOR
STATUS
approved