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A212182 Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists exponents of distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).

%I A212182 #28 Nov 04 2018 00:24:27
%S A212182 0,1,2,1,1,2,1,3,1,2,2,4,1,2,1,1,3,1,1,2,2,1,4,1,1,3,2,1,4,2,1,3,1,1,
%T A212182 1,2,2,1,1,4,1,1,1,3,2,1,1,4,2,1,1,3,3,1,1,5,2,1,1,4,3,1,1,6,2,1,1,4,
%U A212182 2,2,1,3,2,1,1,1,4,4,1,1,5,2,2,1,4,2,1
%N A212182 Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists exponents of distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).
%C A212182 Length of row n = A108602(n).
%C A212182 For n > 1, row n of table gives the "nonincreasing order" version of the prime signature of A002182(n) (cf. A212171). This order is also the natural order of the exponents in the prime factorization of any highly composite number.
%C A212182 The distinct prime factors corresponding to exponents in row n are given by A318490(n, k), where k = 1, 2, 3, ..., A108602(n).
%D A212182 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 A212182 Peter J. Marko, <a href="/A212182/b212182.txt">Table of i, a(i) for i = 1..10022</a> (corresponding to first n = 584 rows of irregular triangle; using data from Flammenkamp)
%H A212182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly composite numbers</a>
%H A212182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>
%H A212182 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/HCN.bz2">List of the first 779,674 highly composite numbers</a>
%H A212182 Peter J. Marko, <a href="/A212182/a212182.txt">Table of n, T(n, k) by rows for n = 1..10000</a> (using data from Flammenkamp)
%H A212182 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper15/page1.htm">Highly Composite Numbers</a>
%F A212182 Row n equals row A002182(n) of table A124010. For n > 1, row n equals row A002182(n) of table A212171.
%e A212182 First rows read:
%e A212182   0;
%e A212182   1;
%e A212182   2;
%e A212182   1, 1;
%e A212182   2, 1;
%e A212182   3, 1;
%e A212182   2, 2;
%e A212182   4, 1;
%e A212182   2, 1, 1;
%e A212182   3, 1, 1;
%e A212182   2, 2, 1;
%e A212182   4, 1, 1;
%e A212182   ...
%e A212182 1st row: A002182(1) = 1 so T(1, 1) = 0;
%e A212182 2nd row: A002182(2) = 2^1 so T(2, 1) = 1;
%e A212182 3rd row: A002182(3) = 4 = 2^2 so T(3, 1) = 2;
%e A212182 4th row: A002182(4) = 6 = 2^1 * 3^1 so T(4, 1) = 1 and T(4, 2) = 1;
%e A212182 5th row: A002182(5) = 12 = 2^2 * 3^1 so T(5, 1) = 2 and T(5, 2) = 1;
%e A212182 6th row: A002182(6) = 24 = 2^3 * 3^1 so T(6, 1) = 3 and T(6, 2) = 1.
%Y A212182 Row n has length A108602(n), n >= 2.
%Y A212182 Cf. A000040, A002182, A124010, A212171, A318490.
%K A212182 nonn,tabf
%O A212182 1,3
%A A212182 _Matthew Vandermast_, Jun 08 2012
%E A212182 Edited by _Peter J. Marko_, Aug 30 2018