OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
FORMULA
a(n) = A007947(A053165(n)) = A053166(A053165(n)) = n/(A053164(n)*A000190(n)) = A053166(n)/A053164(n) = A056553(n)^(1/4).
If n = Product_{j} Pj^Ej then a(n) = Product_{j} Pj^Fj, where Fj = 0 if Ej is 0 or a multiple of 4 and Fj = 1 otherwise.
Multiplicative with a(p^e) = p^(if 4|e, then 0, else 1). - Mitch Harris, Apr 19 2005
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(8)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.3513111135... . - Amiram Eldar, Oct 27 2022
EXAMPLE
a(64) = 2 because 4th-power-free part of 64 is 4 and power-free kernel of 4 is 2.
MATHEMATICA
f[p_, e_] := p^If[Divisible[e, 4], 0, 1]; a[n_] := Times @@ (f @@@ FactorInteger[ n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)
PROG
(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, if (frac(f[k, 2]/4), f[k, 2] = 1, f[k, 2] = 0)); factorback(f); \\ Michel Marcus, Feb 28 2019
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Henry Bottomley, Jun 25 2000
STATUS
approved