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%I #11 Jul 08 2019 12:29:38
%S 1,2,2,1,3,2,1,2,1,2,1,2,3,4,1,1,2,3,4,1,2,3,2,1,1,1,2,2,1,2,3,2,1,4,
%T 1,2,4,1,1,2,3,2,1,4,1,2,4,1,2,2,3,1,1,6,1,2,1,2,2,1,2,2,3,1,1,6,3,2,
%U 1,4,1,2,1,2,2,3,1,6,3,2,4,1,1,1,1,2,2
%N a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.
%C Also, largest number k such that, for each integer j from 1 to k, more multiples of j appear among the divisors of A002182(n) than appear among the divisors of any smaller positive integer.
%C For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k = n/j is highly composite (A002182).
%H Amiram Eldar, <a href="/A181810/b181810.txt">Table of n, a(n) for n = 1..10000</a>
%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly composite number</a>
%e 360 is a member of A002182, twice a member of A002182 (360/2 = 180), and three times a member of A002182 (360/3 = 120), but is not four times a member of A002182 (360/4 = 90 is not a member of A002182). Since A002182(13) = 360, a(13) = 3.
%e 360 also sets records for the number of its divisors, the number of its divisors that are multiples of 2 (cf. A181808), and the number of its divisors that are multiples of 3, but not the number of its divisors that are multiples of 4.
%t f[hc_, n_] := Module[{k=1}, While[MemberQ[hc, n/k], k++]; k-1]; s={}; hc={}; dm = 0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[hc, n]]; AppendTo[s, f[hc, n]], {n, 1, 10^5}]; s (* _Amiram Eldar_, Jul 08 2019 *)
%Y a(n) equals the largest number k such that each number from 1 to k appears in row A002182(n) of A181803. a(n) also equals the largest number k such that each of the first k members of row A002182(n) of A056538 is highly composite.
%Y See also A181801, A181808, A181809.
%K nonn
%O 1,2
%A _Matthew Vandermast_, Nov 27 2010
%E a(5) corrected and more terms added by _Amiram Eldar_, Jul 08 2019
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