OFFSET
1,2
COMMENTS
The generalized k-super abundant numbers are those such that sigma_k(n)/(n^k) > sigma_k(m)/(m^k) for all m < n, where sigma_k(n) is the sum of the k-th powers of the divisors of n.
Pillai called these numbers "highly abundant numbers of the 2nd order". - Amiram Eldar, Jun 30 2019
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..251
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
Srinivasa Ramanujan, Highly composite numbers, Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal, Vol. 1, No. 2 (1997), pp. 119-153, alternative link.
FORMULA
EXAMPLE
For i=24, sigma_2(24)/(24^2)=850/576=1.47569, a new record, thus 24 is in the sequence.
MATHEMATICA
s = {1}; a = 1; Do[ If[DivisorSigma[2, n]/(n^2) > a, a = DivisorSigma[2, n]/(n^2); AppendTo[s, n]], {n, 10000000}]; s
CROSSREFS
KEYWORD
nonn
AUTHOR
Ben Branman, Mar 01 2012
EXTENSIONS
More terms from Amiram Eldar, May 12 2019
STATUS
approved