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).

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, 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, 2, 6, 4, 5, 8, 3, 6, 6, 7, 4, 3, 6, 4, 9, 2, 3, 2, 3, 0, 0, 3

Offset: 1

Author

Frank M Jackson, May 07 2012

Keywords

• nonn
• cons

Comments

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.

Examples

			1.0388205776091298930081555627382465269336112084545034825058980...
		

Links

• J. Barkley Rosser, Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962 64-94
• Eric Weisstein, Chebyshev Functions.

Crossrefs

Cf. A003418

Programs

Mathematica

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]

Formula

C = log(955888052326228459513511038256280353796626534577600)/113