A309015 - OEIS

%I #13 Jul 13 2019 04:04:45

%S 1,2,4,6,12,24,36,48,72,144,216,288,432,576,864,1296,1728,2592,3456,

%T 5184,6912,10368,15552,20736,31104,41472,62208,82944,93312,124416,

%U 186624,248832,373248,497664,746496,995328,1119744,1492992,2239488,2985984,4478976,5971968

%N 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).

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

%C 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).

%C 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.

%H Amiram Eldar, <a href="/A309015/b309015.txt">Table of n, a(n) for n = 1..10000</a>

%H Pascal Alessandri and Valérie Berthé, <a href="https://www.theoremoftheday.org/Docs/3dAlessandriBerthe.pdf">Three distance theorems and combinatorics on words</a>, Enseignement Mathématique, Vol. 44 (1998), pp. 103-132.

%H Gérard Bessi, <a href="https://eudml.org/doc/181966">Etude des nombres 2-hautement composés</a>, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.

%H Gérard Bessi and J. L. Nicolas, <a href="http://math.univ-lyon1.fr/homes-www/nicolas/bessi2hc.pdf">Nombres 2-hautement composés</a>, J. Math. pures et appl., Vol. 56 (1977), pp. 307-326.

%H Amiram Eldar, <a href="/A309015/a309015.txt">Table of n, a(n), A000005(a(n)) for n = 1..10000</a>

%H Jean-Louis Nicolas, <a href="http://math.univ-lyon1.fr/homes-www/nicolas/openquestions.pdf">Some open questions</a>, The Ramanujan Journal, Vol. 9 (2005), pp. 251-264.

%t 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 *)

%t 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 *)

%Y Cf. A000005, A002182, A003586, A070915.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 06 2019