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%I #7 Mar 30 2012 17:27:19
%S 1,1,2,3,1,2,4,5,1,3,6,7,2,4,8,9,5,10,11,1,2,3,6,12,13,7,14,15,4,8,16,
%T 17,3,9,18,19,5,10,20,21,11,22,23,1,2,4,6,12,24,25,13,26,27,7,14,28,
%U 29,5,15,30,31,8,16,32,33,17,34,35,1,3,6,9,18,36,37,19,38,39,10,20,40,41,7,21,42
%N Triangle read by rows: T(n,k) is the k-th smallest divisor d of n such that n sets a record for the number of its divisors that are multiples of d.
%C In other words, row n contains a particular divisor d of n iff more multiples of d appear among the divisors of n than appear among the divisors of any smaller positive integer. Cf. A181808.
%C Row n contains A181801(n) numbers, the largest of which is n. T(n,k) * A180802(n, A181801(n)-k+1) = n.
%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, j appears in row n iff k=n/j is a member of A002182.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly composite number</a>
%F T(n,k) = n/(A180802(n, A181801(n)-k+1)).
%e First rows read: 1; 1,2; 3; 1,2,4; 5; 1,3,6; 7; 2,4,8; 9; 5,10; 11; 1,2,3,6,12;...
%e 6 has four divisors (1, 2, 3 and 6). Of those divisors, 1, 3 and 6 appear in row 6.
%e a. The divisors of 6 include four multiples of 1 (1, 2, 3 and 6); two multiples of 3 (3 and 6), and one multiple of 6 (6). No positive integer smaller than 6 has more than three multiples of 1 among its divisors; hence, 1 appears in row 6. Also, no positive integer smaller than 6 has more than one multiple of 3 among its divisors, or has any multiple of 6 among its divisors. Hence, 3 and 6 both appear in row 6.
%e b. On the other hand, although 6 includes two multiples of 2 among its divisors (2 and 6), so does a smaller positive integer (4, whose even divisors are 2 and 4). Accordingly, 2 is not included in row 6.
%e The divisors of 6 that appear in row 6 are therefore 1, 3 and 6. Note that 1, 3 and 6 equal 6/6, 6/2 and 6/1 respectively, and all of the denominators in those fractions are highly composite numbers (A002182).
%Y For the highly composite divisors of n, see A181802. See also A181808, A181809, A181810.
%K nonn,tabf
%O 1,3
%A _Matthew Vandermast_, Nov 27 2010
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