A056554 Powerfree kernel of 4th-powerfree part of n.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 1, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 3, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38

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

Cf. A000190, A007947, A008835, A013666, A053164, A053165, A053166, A053167, A056552, A056553, A056555.

Sequence in context: A086297 A261969 A281495 * A346486 A088835 A007947

Adjacent sequences: A056551 A056552 A056553 * A056555 A056556 A056557

Keyword

nonn,mult

Author

Henry Bottomley, Jun 25 2000

Status

approved