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).

1, 2, 4, 6, 8, 12, 24, 36, 48, 60, 72, 96, 120, 180, 240, 360, 480, 720, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 151200, 166320, 221760

Offset: 1

Author

Amiram Eldar, Jun 30 2019

Keywords

• nonn

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.

Crossrefs

Cf. A002182, A007425.

Programs

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