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A166981 Superabundant numbers (A004394) that are highly composite (A002182).

%I A166981 #23 May 09 2021 13:33:48
%S A166981 1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,
%T A166981 10080,15120,25200,27720,55440,110880,166320,277200,332640,554400,
%U A166981 665280,720720,1441440,2162160,3603600,4324320,7207200,8648640,10810800
%N A166981 Superabundant numbers (A004394) that are highly composite (A002182).
%C A166981 The intersection of superabundant and highly composite numbers has exactly 449 terms, the largest of which is 2^10 * 3^6 * 5^4 * 7^3 * 11^3 * 13^2 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37*...*347.
%C A166981 The argument showing that this is a finite sequence seems to be given in A166735. - _N. J. A. Sloane_, Jan 04 2019
%C A166981 Pillai proved that this sequence is finite and asked for its number of terms (he used the term "highly abundant" for superabundant numbers). - _Amiram Eldar_, Jun 30 2019
%C A166981 From _Michael De Vlieger_, Dec 29 2020: (Start)
%C A166981 All terms are products of primorials (numbers in A002110), thus, all terms are also in A025487, itself a subsequence of A055932.
%C A166981 Since the colossally abundant numbers (CA, A004490) are also superabundant, and since the superior highly composite (SHC A002201) numbers are also highly composite, the finite sequence A224078 containing numbers both CA and SHC is a subsequence of this sequence. Likewise, A304234 (numbers that are SA, HC, & SHC but not CA) and A304235 (numbers that are SA, HC, & CA but not SHC), and A338786 (SA and HC, but neither CA nor SHC) are mutually exclusive finite subsequences of this sequence. (End)
%H A166981 T. D. Noe, <a href="/A166981/b166981.txt">Table of n, a(n) for n = 1..449</a> (complete sequence)
%H A166981 Michael De Vlieger, <a href="/A166981/a166981.png">Annotated plot of a(n)</a> at (x,y) = (a(n)/P), P) where P = A002110(A001221(a(n)) showing all 449 terms.
%H A166981 Thomas Fink, <a href="https://arxiv.org/abs/1912.07979">Recursively divisible numbers</a>, arXiv:1912.07979 [math.NT], 2019. Mentions this sequence.
%H A166981 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 A166981 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.
%Y A166981 Cf. A002110, A002182, A004394, A025487, A055932, A166735 (SA numbers that are not HC numbers), A224078, A304234, A304235, A308913, A338786.
%K A166981 fini,full,nonn
%O A166981 1,2
%A A166981 _T. D. Noe_, Oct 26 2009