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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
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
1,9
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.
LINKS
David W. Wilson, Table of n, a(n) for n = 1..820 (Rows n=1..40 of triangle, flattened).
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
EXAMPLE
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),
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
David W. Wilson, Apr 02 2012
EXTENSIONS
Edited by Peter Munn, May 22 2025
STATUS
approved