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A206431 Decimal expansion of constant C = maximum value that psi(n)/n reaches where psi(n)=log(lcm(1,2,...,n)) and lcm(1,2,...,n)=A003418(n).

%I A206431 #17 Feb 16 2025 08:33:16
%S A206431 1,0,3,8,8,2,0,5,7,7,6,0,9,1,2,9,8,9,3,0,0,8,1,5,5,5,6,2,7,3,8,2,4,6,
%T A206431 5,2,6,9,3,3,6,1,1,2,0,8,4,5,4,5,0,3,4,8,2,5,0,5,8,9,8,0,3,0,3,8,2,4,
%U A206431 2,6,4,5,8,3,6,6,7,4,3,6,4,9,2,3,2,3,0,0,3
%N A206431 Decimal expansion of constant C = maximum value that psi(n)/n reaches where psi(n)=log(lcm(1,2,...,n)) and lcm(1,2,...,n)=A003418(n).
%C A206431 According to Rosser and Schoenfeld (1961), the second Chebyshev function psi(n)=log(lcm(1,2,...,n)) ~ n. Consequently, the function log(lcm(1,2,...,n))/n tends to 1 as n tends to infinity, however it has a maximum value of 1.03882... when n=113. In precise terms this constant is log(955888052326228459513511038256280353796626534577600)/113 and it provides an upper bound for log(lcm(1,2,...,n)) <= log(955888052326228459513511038256280353796626534577600)/113*n for all n>0.
%H A206431 J. Barkley Rosser, Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Illinois J. Math. 6 1962 64-94
%H A206431 Eric Weisstein, <a href="https://mathworld.wolfram.com/ChebyshevFunctions.html">Chebyshev Functions</a>.
%F A206431 C = log(955888052326228459513511038256280353796626534577600)/113
%e A206431 1.0388205776091298930081555627382465269336112084545034825058980...
%t A206431 table=Table[Log[LCM @@ Range[n]]/n, {n, 1, 1000}]; max=Max[table]; n=1; While[table[[n]]!=max, n++]; Print[N[max, 100]," at n = ",n]
%Y A206431 Cf. A003418
%K A206431 nonn,cons
%O A206431 1,3
%A A206431 _Frank M Jackson_, May 07 2012