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A309015
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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).
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3
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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