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

1, 5, 13, 71, 97, 1355, 793, 19163, 53473, 292355, 60073, 13102907, 535537, 78584915, 790859641, 3523099499, 43112257, 99646519235, 387682633, 2764285630427, 7604811750289, 7337148996275, 31385253913, 2226944658077771, 3656440886376673, 2341258386360995, 80539587570991081

Offset: 1

Author

Amiram Eldar, Mar 29 2025

Keywords

• nonn
• easy
• frac

Comments

The denominator that corresponds to a(n) is 3*6^(n-1) = A169604(n-1) = A081341(n).

Examples

			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, ...
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).
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007310, A072078, A081341 (denominators), A169604, A171126.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(n-#) * 3^(n-n/#) &]; Array[a, 30]

PARI

a(n) = sumdiv(n, d, 2^(n-d)*3^(n-n/d));

Formula

a(n) = Sum_{d|n} 2^(n-d) * 3^(n-n/d).

a(p) = 2^(p-1) + 3^(p-1).

Let f(n) = a(n)/A081341(n). Then:

f(n) = (1/3) * Sum_{d|n} (1/2)^(d-1) * (1/3)^(n/d-1).

Sum_{n>=1} f(n) = 1.

Sum_{n>=1} n * f(n) = 3 (the asymptotic mean of A072078).

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.