A076633 Incorrect guess for index of n-th local maxima (in decreasing order) of f(k) = (sigma(k) - H_k)/(exp(H_k)log(H_k)), where H_k = 1 + 1/2 + 1/3 + ... + 1/k.

12, 120, 60, 2520, 5040, 360, 24, 840, 55440, 10080

Offset: 1

Author

Luke Pebody (pebodyl(AT)msci.memphis.edu), Oct 22 2002

Keywords

• nonn

Comments

Lagarias showed that the Riemann Hypothesis is equivalent to the formula sigma(k) <= H_k + exp(H_k)log(H_k) for all k >= 1 with equality only when k=1. In other words f(k)<1 for all k. At first glance it seems that f(12) is the largest value of f, followed by f(120), f(60) and so on. Proving that f(12) is indeed the largest value would prove the Riemann Hypothesis. However, f(12) is not the largest value.

The terms shown are merely the maxima for "small" values of k. If the function f(k) is evaluated at colossally abundant numbers (A004490), we find that beyond the 58th colossally abundant number, which is over 10^76, the function is greater than f(12) and increasing at each subsequence colossally abundant number. Use A073751 to generate colossally abundant numbers not in A004490. - T. D. Noe, Oct 24 2002

Links

• J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
• T. D. Noe, Plot of the function f for the first 200 colossally abundant numbers

Crossrefs

Cf. A057640, A057641, A004490, A073751.