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A181803
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.
7
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, 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, 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
OFFSET
1,3
COMMENTS
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.
Row n contains A181801(n) numbers, the largest of which is n. T(n,k) * A180802(n, A181801(n)-k+1) = n.
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.
LINKS
Eric Weisstein's World of Mathematics, Highly composite number
FORMULA
T(n,k) = n/(A180802(n, A181801(n)-k+1)).
EXAMPLE
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;...
6 has four divisors (1, 2, 3 and 6). Of those divisors, 1, 3 and 6 appear in row 6.
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.
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.
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).
CROSSREFS
For the highly composite divisors of n, see A181802. See also A181808, A181809, A181810.
Sequence in context: A341826 A372479 A257669 * A144962 A227542 A338900
KEYWORD
nonn,tabf
AUTHOR
Matthew Vandermast, Nov 27 2010
STATUS
approved