OFFSET
1,2
COMMENTS
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.
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).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..603
Amiram Eldar, Table of n, a(n), d_3(a(n)) for n = 1..603
Jean-Louis Nicolas, On highly composite numbers, 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.
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.
S. Sivasankaranarayana Pillai, Highly Composite Numbers of the t th Order, J. Indian Math. Soc., Vol. 8 (1944), pp. 61-74.
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 30 2019
STATUS
approved