|
|
A342477
|
|
The squarefree part of the powerful numbers: a(n) = A007913(A001694(n)).
|
|
1
|
|
|
1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 3, 1, 5, 2, 1, 1, 1, 2, 6, 1, 3, 1, 2, 1, 1, 7, 1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 6, 1, 1, 2, 3, 10, 1, 1, 5, 2, 1, 1, 1, 3, 11, 2, 1, 7, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 6, 5, 1, 2, 1, 3, 13, 1, 1, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
FORMULA
|
Theorems 1-4 by Copil and Panaitopol (2007):
Sum_{k=1..n} a(k) ~ (3/Pi^2)*zeta(4/3)*n^(3/2) + O(sqrt(n)*log(n)).
Sum_{k=1..n} a(k)^2 ~ n/3 + O(n^(5/6)).
Sum_{k=1..n} 1/a(k) ~ (zeta(5/2)/zeta(5))*sqrt(n) + O(log(n)).
Product_{k=1..n} a(k) ~ exp(c*sqrt(n) + O(n^(1/3)*log(n))), where c = Sum_{k>=1} f(A005117(k)), and f(x) = log(x)/x^(3/2).
|
|
MATHEMATICA
|
f[p_, e_] := p^Mod[e, 2]; sqfp[n_] := Times @@ f @@@ FactorInteger[n]; powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; sqfp /@ Select[Range[1000], powQ]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|