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%I #14 Apr 28 2016 12:42:07
%S 0,0,2,0,2,3,2,2,4,2,3,2,2,4,3,2,4,2,3,2,2,4,3,2,4,2,3,3,5,2,4,3,6,2,
%T 4,2,2,3,2,4,4,5,2,2,4,2,3,3,5,2,4,3,6,2,4,2,2,5,3,4,4,5,2,2,6,3,4,2,
%U 3,3,5,2,4,3,6,2,4,2,2,5,3,4,4,5,2,2,6
%N Row n of table gives exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.
%C Length of row n equals A212185(n) if A212185(n) is positive, or 1 if A212185(n) = 0.
%C Row n of table represents second signature of A002182(n) (cf. A212172). The use of 0 in the table to represent squarefree highly composite numbers accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.
%C No row is repeated an infinite number of times in the table. The contrary to this would imply that at least one integer appeared in A212183 an infinite number of times - something that Ramanujan proved to be false (cf. Ramanujan link). It would be interesting to know if there is an upper bound on the number of times a row can appear.
%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 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/page34.htm">Highly Composite Numbers</a> (p. 34)
%F Row n is identical to row A002182(n) of table A212172.
%e First rows read: 0; 0; 2; 0; 2; 3; 2,2; 4; 2; 3; 2,2; 4;...
%e 12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Only exponents that are 2 or greater appear in a number's second signature; therefore, 12's second signature is {2}. Since 12 = A002182(5), row 5 represents the second signature {2}.
%Y Cf. A002182, A212172, A212182, A212183, A212185.
%K nonn,tabf
%O 1,3
%A _Matthew Vandermast_, Jul 01 2012
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