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%I #28 Sep 25 2022 04:28:09
%S 1,2,4,6,12,24,48,60,120,240,360,720,840,1680,2520,5040,10080,15120,
%T 25200,27720,55440,110880,166320,277200,332640,360360,720720,1441440,
%U 2162160,3603600,4324320,7207200,10810800,12252240,21621600,24504480,36756720,61261200
%N Generalized 2-super abundant numbers.
%C 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.
%C 1-super abundant numbers are A004394. 0-super abundant numbers are A002182.
%C Pillai called these numbers "highly abundant numbers of the 2nd order". - _Amiram Eldar_, Jun 30 2019
%H Amiram Eldar, <a href="/A208767/b208767.txt">Table of n, a(n) for n = 1..251</a>
%H S. Sivasankaranarayana Pillai, <a href="https://web.archive.org/web/20150912090449/http://www.calmathsoc.org/bulletin/article.php?ID=B.1943.35.20">Highly abundant numbers</a>, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
%H S. Sivasankaranarayana Pillai, <a href="https://archive.org/details/in.ernet.dli.2015.282686/page/n825">On numbers analogous to highly composite numbers of Ramanujan</a>, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
%H Srinivasa Ramanujan, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly composite numbers</a>, Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal, Vol. 1, No. 2 (1997), pp. 119-153, <a href="http://math.univ-lyon1.fr/homes-www/nicolas/ramanujanNR.pdf">alternative link</a>.
%F Limit_{n->oo} A001157(a(n))/a(n)^2 = zeta(2) (A013661). - _Amiram Eldar_, Sep 25 2022
%e For i=24, sigma_2(24)/(24^2)=850/576=1.47569, a new record, thus 24 is in the sequence.
%t s = {1}; a = 1; Do[ If[DivisorSigma[2, n]/(n^2) > a, a = DivisorSigma[2, n]/(n^2); AppendTo[s, n]], {n, 10000000}]; s
%Y Cf. A002182, A004394, A004490, A002201, A001157, A013661, A017667, A017668.
%Y Subsequence of A025487.
%K nonn
%O 1,2
%A _Ben Branman_, Mar 01 2012
%E More terms from _Amiram Eldar_, May 12 2019