OFFSET
1,2
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 06 2019
STATUS
approved