1,2

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.

N. G. de Bruijn, "On the number of uncancelled elements in the sieve of Eratosthenes", Proc. Neder. Akad. Wetensch, 1950

A. A. Buchstab, "Asymptotic estimates of a general number-theoretic function", Mat. Sbornik 44 (1937), 1239-1246.

M. F. Hasler, Table of n, a(n) for n = 1..36

"Counting Integers and their Multiplicities" on mersenneforum.org

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

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);

(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));

Cf. A101459.

Cf. A027424.

Adjacent sequences: A126956 A126957 A126958 this_sequence A126960 A126961 A126962

Sequence in context: A020030 A121393 A003316 this_sequence A058861 A105668 A064856

nonn

M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Mar 19 2007, Mar 22 2007