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 A126959 a(k) = k! * lim_{n->oo} card({ i*j; i=1..k, j=1..n })/n. 2
 1, 3, 12, 58, 352, 2376, 19296, 168912, 1670976, 18000000, 219916800, 2781561600, 39605760000, 584889984000, 9253091635200, 154909552896000, 2834240274432000, 52918877491200000, 1074184895250432000 (list; graph; refs; listen; history; text; internal format)
 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. "Counting Integers and their Multiplicities" on mersenneforum.org 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 Cf. A101459, A027424. Sequence in context: A121393 A003316 A298419 * A181328 A058861 A105668 Adjacent sequences: A126956 A126957 A126958 * A126960 A126961 A126962 KEYWORD nonn AUTHOR M. F. Hasler, Mar 19 2007, Mar 22 2007 STATUS approved

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Last modified October 4 19:04 EDT 2023. Contains 365888 sequences. (Running on oeis4.)