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 A181930 Table T(d,n), where T(d,n)/lcm(1..d) gives the probability that d is the n-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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS By probability is meant limit density on [1,n] as n grows without bound. Equivalently: T(n,d) is lcm(1..d) times the asymptotic density of the set of natural numbers whose n-th divisor is equal to d. Rows are (1), (0,1), (0,1,1), (0,0,2,1), .... LINKS David W. Wilson, Table of n, a(n) for n = 1..820 FORMULA T(d,d) = 1. T(d,n) = 0 if n < tau(d) = A000005(d). If d is a divisor of n, then every divisor of d is also a divisor of n, and d is therefore at least the tau(n)-th divisor of n. T(d,n) > 0 for all tau(d) <= n <= d. To show this, let S be the set of the divisors of d along with the smallest n-tau(d) non-divisors of d. Then S has n elements, the largest being d. The elements of S are the smallest divisors of lcm(S), so d is the n-th divisor of lcm(S). Hence every number of the form lcm(S) + k*lcm(1..d) has n-th divisor d, so numbers with n-th divisor d have asymptotic density >= 1/lcm(1..d), and T(d,n) > 0. Sum_{d>=1} T(d,n)/lcm(1..d) = 1. Sum_{n=1..d} T(d,n)/lcm(1..d) = 1/d. T(d,tau(d)) = (lcm(1..d)/d) * Product( (q-1)/q, q prime and there is an a with q^a < d and q^a does not divide d). In particular, if p is prime, then T(p,2) = (lcm(1..p)/p) * product( (q-1)/q, q prime and q < d). - Benoit Jubin, Apr 02 2012 EXAMPLE T(5,4) = 4th element of the fifth row (0,4,4,3,1) = 3. T(5,4)/lcm(1..5) = T(5,4)/A003418(5) = T(5,4)/60 = 1/20 is the probability that 5 is the 4th divisor of an integer. CROSSREFS Sequence in context: A049243 A077908 A052922 * A256797 A109167 A066426 Adjacent sequences:  A181927 A181928 A181929 * A181931 A181932 A181933 KEYWORD nonn,tabl AUTHOR David W. Wilson, Apr 02 2012 STATUS approved

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Last modified May 17 14:50 EDT 2021. Contains 343972 sequences. (Running on oeis4.)