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Number of exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)).
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%I #11 Jun 30 2019 10:42:26

%S 0,0,1,0,1,1,2,1,1,1,2,1,2,2,1,2,1,2,2,2,2,2,2,3,2,2,3,2,2,2,2,2,3,2,

%T 2,3,2,2,2,2,2,2,3,2,2,3,2,3,3,2,3,3,2,3,3,2,2,3,2,3,3,2,3,3,2,3,3,2,

%U 2,3,3,2,3,3,2,3,3,3,3,3,3,4,2,4,3,4,2

%N Number of exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)).

%C Length of row n of A212184 equals a(n) if a(n) is positive, 1 otherwise.

%D S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

%H Amiram Eldar, <a href="/A212185/b212185.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 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page1.htm">Highly Composite Numbers</a>

%e The canonical prime factorization of 720 (2^4*3^2*5) has 2 exponents that equal or exceed 2. Since 720 = A002182(14), a(14) = 2.

%t s={}; dm=0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; e = FactorInteger[n][[;;,2]]; AppendTo[s, Count[e, _?(# > 1 &)]]], {n, 1, 10^6}]; s (* _Amiram Eldar_, Jun 30 2019 *)

%Y Cf. A002182, A212172, A212182, A212184.

%K nonn

%O 1,7

%A _Matthew Vandermast_, Jul 16 2012