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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.

%I A181930 #39 May 22 2025 21:02:26
%S A181930 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,
%T A181930 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,
%U A181930 720,392,384,188,154,70,26,9,1,0,0,0,0,0,480,848,560
%N 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.
%C A181930 By probability is meant limit density on [1,n] as n grows without bound.
%C A181930 Equivalently, T(d,k) is lcm(1..d) times the asymptotic density of the numbers whose k-th divisor is d.
%H A181930 David W. Wilson, <a href="/A181930/b181930.txt">Table of n, a(n) for n = 1..820</a> (Rows n=1..40 of triangle, flattened).
%F A181930 T(d,d) = 1.
%F A181930 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.)
%F A181930 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]
%F A181930 Sum_{d>=k} T(d,k)/lcm(1..d) = 1.
%F A181930 Sum_{k=1..d} T(d,k)/lcm(1..d) = 1/d.
%F A181930 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
%e A181930 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.
%e A181930 Triangle begins:
%e A181930   (1),
%e A181930   (0,1),
%e A181930   (0,1,1),
%e A181930   (0,0,2,1),
%e A181930   (0,4,4,3,1),
%e A181930   ...
%Y A181930 Cf. A000005, A003418, A027750, A281890.
%K A181930 nonn,tabl
%O A181930 1,9
%A A181930 _David W. Wilson_, Apr 02 2012
%E A181930 Edited by _Peter Munn_, May 22 2025