A309015 2-highly composite numbers: 3-smooth numbers (A003586) k with d(k) > d(j) for all 3-smooth numbers j < k, where d(k) is the number of divisors of k (A000005).

1, 2, 4, 6, 12, 24, 36, 48, 72, 144, 216, 288, 432, 576, 864, 1296, 1728, 2592, 3456, 5184, 6912, 10368, 15552, 20736, 31104, 41472, 62208, 82944, 93312, 124416, 186624, 248832, 373248, 497664, 746496, 995328, 1119744, 1492992, 2239488, 2985984, 4478976, 5971968

Offset: 1

Author

Amiram Eldar, Jul 06 2019

Keywords

• nonn

Comments

Also numbers with record numbers of divisors among the numbers with at most 2 distinct prime factors (A070915).

Bessi and Nicolas proved that there exists a constant c such that the number of 2-highly composite numbers smaller than x is larger than c*(log(x))^(4/3).

In general, k-highly composite numbers (defined by Nicolas, 2005) are numbers with a record number of divisors where only p(k)-smooth numbers are being considered. Equivalently only numbers with at most k distinct prime factors can be considered.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Pascal Alessandri and Valérie Berthé, Three distance theorems and combinatorics on words, Enseignement Mathématique, Vol. 44 (1998), pp. 103-132.
• Gérard Bessi, Etude des nombres 2-hautement composés, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.
• Gérard Bessi and J. L. Nicolas, Nombres 2-hautement composés, J. Math. pures et appl., Vol. 56 (1977), pp. 307-326.
• Amiram Eldar, Table of n, a(n), A000005(a(n)) for n = 1..10000
• Jean-Louis Nicolas, Some open questions, The Ramanujan Journal, Vol. 9 (2005), pp. 251-264.

Crossrefs

Cf. A000005, A002182, A003586, A070915.

Programs

Mathematica

dmax = 0; s = {}; Do[If[EulerPhi[6n] == 2n, d = DivisorSigma[0, n]; If[d > dmax, dmax = d; AppendTo[s, n]]], {n, 1, 10^4}]; s (* after Artur Jasinski at A003586 *)
Block[{n = 10^7, s, t}, s = Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2@ n}, {b, 0, Log[3, n/(2^a)]}]; t = DivisorSigma[0, s]; Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Jul 09 2019 *)