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A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.

%I A382491 #7 Mar 29 2025 04:23:13
%S A382491 1,5,13,71,97,1355,793,19163,53473,292355,60073,13102907,535537,
%T A382491 78584915,790859641,3523099499,43112257,99646519235,387682633,
%U A382491 2764285630427,7604811750289,7337148996275,31385253913,2226944658077771,3656440886376673,2341258386360995,80539587570991081
%N A382491 a(n) is the numerator of the asymptotic density of the numbers whose number of 3-smooth divisors is n.
%C A382491 The denominator that corresponds to a(n) is 3*6^(n-1) = A169604(n-1) = A081341(n).
%H A382491 Amiram Eldar, <a href="/A382491/b382491.txt">Table of n, a(n) for n = 1..1000</a>
%F A382491 a(n) = Sum_{d|n} 2^(n-d) * 3^(n-n/d).
%F A382491 a(p) = 2^(p-1) + 3^(p-1).
%F A382491 Let f(n) = a(n)/A081341(n). Then:
%F A382491 f(n) = (1/3) * Sum_{d|n} (1/2)^(d-1) * (1/3)^(n/d-1).
%F A382491 Sum_{n>=1} f(n) = 1.
%F A382491 Sum_{n>=1} n * f(n) = 3 (the asymptotic mean of A072078).
%F A382491 Sum_{n>=1} n^2 * f(n) = 18, and therefore, the asymptotic variance of A072078 is 18 - 3^2 = 9, and its asymptotic standard deviation is 3.
%e A382491 Fractions begin with 1/3, 5/18, 13/108, 71/648, 97/3888, 1355/23328, 793/139968, 19163/839808, 53473/5038848, 292355/30233088, 60073/181398528, 13102907/1088391168, ...
%e A382491 a(1) = 1 since a(1)/A081341(1) = 1/3 is the asymptotic density of the numbers with a single 3-smooth divisor, 1, i.e., the numbers that are congruent to 1 or 5 mod 6 (A007310).
%e A382491 a(2) = 5 since a(2)/A081341(2) = 5/18 is the asymptotic density of the numbers with exactly two 3-smooth divisors, either 1 and 2 or 1 and 3, i.e., A171126.
%t A382491 a[n_] := DivisorSum[n, 2^(n-#) * 3^(n-n/#) &]; Array[a, 30]
%o A382491 (PARI) a(n) = sumdiv(n, d, 2^(n-d)*3^(n-n/d));
%Y A382491 Cf. A007310, A072078, A081341 (denominators), A169604, A171126.
%K A382491 nonn,easy,frac
%O A382491 1,2
%A A382491 _Amiram Eldar_, Mar 29 2025