A126959 a(k) = k! * lim_{n->oo} card({ i*j; i=1..k, j=1..n })/n.

1, 3, 12, 58, 352, 2376, 19296, 168912, 1670976, 18000000, 219916800, 2781561600, 39605760000, 584889984000, 9253091635200, 154909552896000, 2834240274432000, 52918877491200000, 1074184895250432000

Offset: 1

Author

M. F. Hasler, Mar 19 2007, Mar 22 2007

Keywords

• nonn

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.

Examples

			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.
		

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.
• "Counting Integers and their Multiplicities" on mersenneforum.org

Crossrefs

Cf. A101459, A027424.

Programs

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);

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));