|
| |
| |
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,16
|
|
|
COMMENTS
|
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...
|
|
|
LINKS
|
|
|
|
FORMULA
|
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).
|
|
|
EXAMPLE
|
The fractions begin with 0, 0, 0, 1/6, 0, 0, 0, 1/4, 1/6, 0, 0, 1/6, ....
|
|
|
MATHEMATICA
|
f[p_, e_] := e/(e + 1); a[1] = 0; a[n_] := Numerator[Plus @@ f @@@ (fct = FactorInteger[n]) - Length[fct]/2]; Array[a, 100]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,frac
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
