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Amiram Eldar, <a href="/A346012/b346012.txt">Table of n, a(n) for n = 1..10000</a>

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f[p_, e_] := e/(e + 1); sa[1] = 0; a[n_] := Numerator[Plus @@ f @@@ (fct = FactorInteger[n]) - Length[fct]/2]; Array[a, 100]

f(n) depends only on the prime signature of n: If n = Product_{i} p_i^e_i, then f(n) = Sum_{i} (e_i - 1)/(2*(e_i + 1)).

Duncan (1961) proved that these two functions have the same average order, log(log(n))/2 , and that their difference has a constant average order, or an asymptotic mean, c (A346011; see the Formula section).

allocated for Amiram Eldara(n) is the numerator of A346009(n)/A346010(n) - A001221(n)/2.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0

1,16

a(n)/A346013 is the difference between two functions, the average number of distinct prime factors of the divisors of n and half the number of distinct prime factors of n.

Duncan (1961) proved that these two functions have the same average order, log(log(n))/2 and that their difference has a constant average order, or an asymptotic mean, c (A346011; see the Formula section).

The nonzero values occur exactly at the nonsquarefree numbers (A013929) and their asymptotic mean is c/(1-6/Pi^2) = 0.2439041253...

R. L. Duncan, <a href="https://www.jstor.org/stable/2311587">Note on the divisors of a number</a>, The American Mathematical Monthly, Vol. 68, No. 4 (1961), pp. 356-359.

Sébastien Gaboury, <a href="http://hdl.handle.net/20.500.11794/18899">Sur les convolutions de fonctions arithmétiques</a>, M.Sc. thesis, Laval University, Quebec, 2007.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Average_order_of_an_arithmetic_function">Average order of an arithmetic function</a>.

Let f(n) = a(n)/A346013(n) be the sequence of fractions. Then:

f(n) depends only on the prime signature of n: If n = Product_{i} p_i^e_i, then f(n) = Sum_{i} (e_i-1)/(2*(e_i+1)).

f(n) = 0 if and only if n is squarefree (A005117), and f(n) > 0 otherwise.

f(n) = (Sum_{p prime, p^2|n} d(n/p^2))/(2*d(n)), where d(n) is the number of divisors of n (A000005).

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} f(n) = 0.095628... (A346011).

The fractions begin with 0, 0, 0, 1/6, 0, 0, 0, 1/4, 1/6, 0, 0, 1/6, ....

f[p_, e_] := e/(e + 1); s[1] = 0; a[n_] := Numerator[Plus @@ f @@@ (fct = FactorInteger[n]) - Length[fct]/2]; Array[a, 100]

Cf. A000005, A001221, A005117, A013929, A346009, A346010, A346011, A346013 (denominators).

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nonn,frac

Amiram Eldar, Jul 01 2021

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