OFFSET
1,2
COMMENTS
a(k) = k! card { i*j, i<=k, j<=k# } / k# where k# = lcm(1,2,3...,k) a(k)/(k+1)! <= 1/2 for all k.
REFERENCES
A. A. Buchstab, "Asymptotic estimates of a general number-theoretic function", Mat. Sbornik 44 (1937), 1239-1246.
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..36
N. G. de Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Proc. Neder. Akad. Wetensch, 1950.
EXAMPLE
a(2)=3/2 since #{ i*j, i=1..2, j=1..2 } / 2 = #{ 1,2, 2,4 } / 2 = #{1,2,4} / 2.
a(3)=2 since #{ i*j, i=1..3, j=1..6 } / 6 = #{ 1,2,3,4,5,6, 2,4,6,8,10,12, 3,6,9,12,15,18 } / 6 = #{ 1,2,3,4,5,6,8,9,10,12,15,18 } / 6.
MAPLE
p:=proc(n) option remember; local s, t, i, j: s:=1; t:={}:
for i from n-1 by -1 to 1+n/(min@op@eval@numtheory[factorset])(n) do
t := t union { ilcm(n, i)/n };
t := select( x-> numtheory[divisors](x) intersect t = { x }, t ):
for j in combinat[powerset](t) do s := s+(-1)^nops(j)/ilcm(op(j)) od:
od; s/n end:
A126959 := k -> k!*add( p(n), n=1..k);
PROG
(PARI) p(n)={ local( cnt=lcm(vector(n-1, j, j)), L=vector(cnt, j, n*j), s=cnt ); forstep( i=n-1, n/factor(n)[1, 1]+1, -1, forstep( j=lcm(n, i)/n, #L, lcm(n, i)/n, if( L[j] && (L[j] % i == 0), L[j]=0; cnt--)); s+=cnt ); s/#L/n } a=vector(16); a[1]=1; for( k=2, #a, a[k]=k*a[k-1]+k!*p(k));
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Mar 19 2007, Mar 22 2007
STATUS
approved