A181930 Triangle T(d,k), where T(d,k)/lcm(1..d) gives the probability that d is the k-th divisor of an integer.

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 4, 4, 3, 1, 0, 0, 0, 4, 5, 1, 0, 16, 20, 12, 6, 5, 1, 0, 0, 0, 48, 20, 26, 10, 1, 0, 0, 96, 40, 52, 44, 36, 11, 1, 0, 0, 0, 72, 48, 66, 34, 22, 9, 1, 0, 576, 720, 392, 384, 188, 154, 70, 26, 9, 1, 0, 0, 0, 0, 0, 480, 848, 560

Offset: 1

Author

David W. Wilson, Apr 02 2012

Keywords

• nonn
• tabl

Comments

By probability is meant limit density on [1,n] as n grows without bound.

Equivalently, T(d,k) is lcm(1..d) times the asymptotic density of the numbers whose k-th divisor is d.

Examples

			T(5,4) = 3. T(5,4)/lcm(1..5) = 3/60 = 1/20 is the probability that 5 is the 4th divisor of an integer.
Triangle begins:
  (1),
  (0,1),
  (0,1,1),
  (0,0,2,1),
  (0,4,4,3,1),
  ...
		

Links

• David W. Wilson, Table of n, a(n) for n = 1..820 (Rows n=1..40 of triangle, flattened).

Crossrefs

Cf. A000005, A003418, A027750, A281890.

Formula

T(d,d) = 1.

T(d,k) = 0 if k < tau(d) = A000005(d). (If d is a divisor of m, then every divisor of d is a divisor of m, and d is therefore at least the tau(d)-th divisor of m.)

T(d,k) > 0 for k with tau(d) <= k <= d. [Appears to have been submitted on basis of a faulty proof. - Peter Munn, May 22 2025]

Sum_{d>=k} T(d,k)/lcm(1..d) = 1.

Sum_{k=1..d} T(d,k)/lcm(1..d) = 1/d.

T(d,tau(d)) = (lcm(1..d)/d) * Product_{q prime and there is an a with q^a < d and q^a does not divide d} (q-1)/q. In particular, if p is prime, then T(p,2) = (lcm(1..p)/p) * Product_{q prime and q < d} (q-1)/q. - Benoit Jubin, Apr 02 2012

Extensions

Edited by Peter Munn, May 22 2025