A084238 a(2) = 1, and for n > 2 a(n) is the least k such that log(k) < k^(1/n) and log(k-1) >= (k-1)^(1/n).

1, 94, 5504, 332106, 24128092, 2099467159, 214910065296, 25438034785805, 3430631121407802, 520643904835474202, 87994213187313363255, 16416338625038083857946, 3355257076845892674934411, 746397968687429806357762425, 179698501514006236611711868382, 46589028541465014633355926255885

Offset: 2

Author

Robert G. Wilson v, May 18 2003

Keywords

• nonn

Comments

A demonstration "that log x increases slower than any power of x. ... No matter how small you make a, the graph of log x is eventually flatter than the graph of x^a."

References

• John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, Washington, D.C., 2003, Page 72 - 75.

Programs

Mathematica

Table[ Floor[ FindRoot[ Log[x]^n == x, {x, 10^(2n)}, AccuracyGoal -> 24, WorkingPrecision -> 34][[1, 2]] + 1], {n, 2, 15}]

Formula

For n = 1, a(n) = 1. For n>=2, a(n) = ceiling(e^(-(n+1)*W-1(-1/(n+1)))) where W-1(x) is the Lambert W function with branch -1. - Joseph C. Y. Wong, Feb 26 2021

Extensions

a(14)-a(16) from Joseph C. Y. Wong, Feb 26 2021

Name clarified by Pontus von Brömssen, Oct 11 2021