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A308912 Highly composite numbers of the 3rd order: numbers k with d_3(k) > d_3(j) for all j < k, where d_3(k) is the 3rd Piltz function, the number of ways of decomposing k into 3 factors (A007425).

%I A308912 #14 Jun 30 2019 14:48:31
%S A308912 1,2,4,6,8,12,24,36,48,60,72,96,120,180,240,360,480,720,1080,1260,
%T A308912 1440,1680,2160,2520,3360,4320,5040,7560,10080,15120,20160,25200,
%U A308912 30240,40320,45360,50400,55440,60480,75600,90720,100800,110880,151200,166320,221760
%N A308912 Highly composite numbers of the 3rd order: numbers k with d_3(k) > d_3(j) for all j < k, where d_3(k) is the 3rd Piltz function, the number of ways of decomposing k into 3 factors (A007425).
%C A308912 Pillai defined highly composite numbers of the t-th order and numbers k with d_t(k) > d_t(j) for all j < k, where d_t(k) is the t-th Piltz function, the number of ways of decomposing k into t factors. The highly composite numbers (A002182) are highly composite numbers of the 2nd order.
%C A308912 The corresponding record values are 1, 3, 6, 9, 10, 18, 30, 36, 45, 54, 60, 63, 90, 108, ... (see the link for more terms).
%H A308912 Amiram Eldar, <a href="/A308912/b308912.txt">Table of n, a(n) for n = 1..603</a>
%H A308912 Amiram Eldar, <a href="/A308912/a308912.txt">Table of n, a(n), d_3(a(n)) for n = 1..603</a>
%H A308912 Jean-Louis Nicolas, <a href="http://math.univ-lyon1.fr/~nicolas/hcnrevisited.pdf">On highly composite numbers</a>, in the book Ramanujan Revisited, Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, 1987, Editors G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan, R.A. Rankin.
%H A308912 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 A308912 S. Sivasankaranarayana Pillai, <a href="http://www.informaticsjournals.com/index.php/jims/article/view/17091">Highly Composite Numbers of the t th Order</a>, J. Indian Math. Soc., Vol. 8 (1944), pp. 61-74.
%t A308912 f[p_, e_] := (e+1)*(e+2)/2;  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; am=0; s={}; Do[a1=a[n]; If[a1 > am, am=a1; AppendTo[s, n]], {n, 1, 100000}]; s
%Y A308912 Cf. A002182, A007425.
%K A308912 nonn
%O A308912 1,2
%A A308912 _Amiram Eldar_, Jun 30 2019