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%I #11 Mar 27 2019 10:10:25
%S 1,2,3,5,6,8,10,12,13,15,14,15,17,16,16,19,17,21,19,20,26,22,25,26,25,
%T 29,28,26,27,28,29,33,33,34,37,37,35,35,39,37,38,38,37,37,38,38,41,38,
%U 37,36,37,37,40,44,44,45,44,44,45,45,49,48,52,51,53,52,51
%N Number of highly composite numbers (m in A002182) in the interval p_k# <= m < p_(k+1)#, where p_i# = A002110(i).
%C Terms m in A002182 (highly composite numbers or HCNs) are products of primes p <= q, where q is the greatest prime factor of m. The primorial A002110(k) is the smallest number that is the product of the k smallest primes. This sequence partitions A002182 using terms in A002110.
%H Michael De Vlieger, <a href="/A307113/b307113.txt">Table of n, a(n) for n = 0..4149</a>
%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">List of the first 779,674 highly composite numbers</a>
%e a(3) = 5 since there are 5 highly composite numbers A002110(3) <= m < A002110(4), i.e., 30 <= m < 210: {36, 48, 60, 120, 180}.
%e n a(n) m such that A002110(n) <= m < A002110(n+1)
%e --------------------------------------------------------------------
%e 0 1 1
%e 1 2 2 4
%e 2 3 6 12 24
%e 3 5 36 48 60 120 180
%e 4 6 240 360 720 840 1260 1680
%e 5 8 2520 5040 7560 10080 15120 20160 25200 27720
%e ...
%t Block[{nn = 8, P, s}, P = Nest[Append[#, #[[-1]] Prime@ Length@ #] &, {1}, nn + 1]; s = DivisorSigma[0, Range@ P[[nn + 1]] ]; s = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Table[Count[s, _?(If[! IntegerQ@ #, 1, #] &@ P[[i]] <= # < P[[i + 1]] &)], {i, nn}]]
%Y Cf. A002110, A002182.
%K nonn
%O 0,2
%A _Michael De Vlieger_, Mar 25 2019
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