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A181805 Number of divisors of A181804(n) that are highly composite (A002182).

%I A181805 #13 Feb 16 2025 08:33:13
%S A181805 1,2,3,3,5,6,6,7,6,7,8,8,8,10,11,14,9,9,12,14,19,15,20,21,21,20,15,22,
%T A181805 22,22,21,23,22,17,23,23,23,24,25,24,25,23,23,25,28,25,27,27,31,22,27,
%U A181805 26,30,18,29,25,32,33,28,29,28,35,25,33,34,31,31,38,37
%N A181805 Number of divisors of A181804(n) that are highly composite (A002182).
%C A181805 a(n) = maximal number of members of A002182 that have a least common multiple of A181804(n). Also, a(n) = length of row A181804(n) in triangles A181802 and A181803.
%C A181805 4, 13 and 16 are the first three positive integers that appear nowhere in this sequence (and, therefore, nowhere in A181801).  It would be interesting to know whether there are others.
%H A181805 Amiram Eldar, <a href="/A181805/b181805.txt">Table of n, a(n) for n = 1..10000</a>
%H A181805 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>.
%H A181805 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly composite number</a>.
%F A181805 a(n) = A181801(A181804(n)).
%e A181805 A181804(10) = 72 has exactly seven divisors that are members of A002182 (namely, 1, 2, 4, 6, 12, 24 and 36). Hence, a(10) = 7.
%t A181805 seq[max_] := Module[{hcn = {}, hcnmax, d, dm = 0, s = {1}}, Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hcn, n]], {n, 1, max}]; hcnmax = hcn[[-1]]; Do[s = Union[Join[s, Select[LCM[hcn[[k]], s], # <= hcnmax &]]], {k, 2, Length[hcn]}]; Do[s[[k]] = Count[hcn, _?(Divisible[s[[k]], #] &)], {k, 1, Length[s]}]; s]; seq[300000] (* _Amiram Eldar_, Jun 23 2023 *)
%Y A181805 A181806(m) is the m-th member of A181804 such that the value of a(n) increases to a record. See also A181807.
%Y A181805 Cf. A002182, A181801, A181802, A181803, A181804.
%K A181805 nonn
%O A181805 1,2
%A A181805 _Matthew Vandermast_, Nov 27 2010
%E A181805 More terms from _Amiram Eldar_, Jun 23 2023