|
%I #40 Oct 08 2018 04:05:47
%S 0,2,2,2,3,2,3,2,3,2,3,2,3,2,3,5,2,3,5,2,3,5,2,3,5,2,3,5,2,3,5,2,3,5,
%T 7,2,3,5,7,2,3,5,7,2,3,5,7,2,3,5,7,2,3,5,7,2,3,5,7,2,3,5,7,2,3,5,7,2,
%U 3,5,7,2,3,5,7,11,2,3,5,7
%N Irregular triangle read by rows T(n,k): T(1,1) = 0; for n > 1, row n lists distinct prime factors of the n-th highly composite number (A002182(n)), where column k = 1, 2, 3, ..., omega(A002182(n)) = A108602(n).
%C The exponents of factors in row n are given by A212182(n).
%H Peter J. Marko, <a href="/A318490/b318490.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 A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly composite numbers</a>
%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.txt">List of the first 1200 highly composite numbers</a>
%H A. Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/HCN.bz2">List of the first 779,674 highly composite numbers</a>
%H Peter J. Marko, <a href="/A318490/a318490_1.txt">Table of n, T(n, k) by rows for n = 1..10000</a> (using data from Flammenkamp)
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram15.html">Highly composite numbers</a>, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.
%e Triangle begins:
%e 0;
%e 2;
%e 2;
%e 2, 3;
%e 2, 3;
%e 2, 3;
%e 2, 3;
%e 2, 3;
%e 2, 3, 5;
%e 2, 3, 5;
%e 2, 3, 5;
%e 2, 3, 5;
%e 2, 3, 5;
%e 2, 3, 5;
%e 2, 3, 5, 7;
%e ...
%e 1st row: A002182(1) = 1 so T(1,1) = 0;
%e 2nd row: A002182(2) = 2 so T(2,1) = 2;
%e 3rd row: A002182(3) = 4 = 2^2 so T(3,1) = 2;
%e 4th row: A002182(4) = 6 = 2 * 3 so T(4,1) = 2 and T(4,2) = 3;
%e 5th row: A002182(5) = 12 = 2^2 * 3 so T(5,1) = 2 and T(5,2) = 3;
%e 6th row: A002182(6) = 24 = 2^3 * 3 so T(6,1) = 2 and T(6,2) = 3.
%Y Row n has length A108602(n), n >= 2.
%Y Cf. A000040, A002182, A212182.
%K nonn,tabf
%O 1,2
%A _Peter J. Marko_, Aug 27 2018
|
