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.
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
Examples
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).
Links
- Eric Weisstein's World of Mathematics, Highly composite number
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