A073753 -id:A073753

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A062378

n divided by largest cubefree factor of n.

+10
10

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

Numerator of n/rad(n)^2, where rad is the squarefree kernel of n (A007947), denominator: A055231. - Reinhard Zumkeller, Dec 10 2002

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

FORMULA

a(n) = n / A007948(n).

a(n) = A003557(A003557(n)). - Antti Karttunen, Nov 28 2017

Multiplicative with a(p^e) = p^max(e-2, 0). - Amiram Eldar, Sep 07 2020

Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Dec 07 2023

MATHEMATICA

f[p_, e_] := p^Max[e-2, 0]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

PROG

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, f[i, 1]^max(f[i, 2]-2, 0)) \\ Charles R Greathouse IV, Aug 08 2013

(Scheme)

(define (A062378 n) (/ n (A007948 n)))

(definec (A007948 n) (if (= 1 n) n (* (expt (A020639 n) (min 2 (A067029 n))) (A007948 (A028234 n)))))

;; Antti Karttunen, Nov 28 2017

CROSSREFS

Cf. A000189, A000578, A007948, A008834, A019555, A048798, A050985, A053149, A053150, A056551, A056552. See A003557 for squares and A062379 for 4th powers.

Differs from A073753 for the first time at n=90, where a(90) = 1, while A073753(90) = 3.

Cf. A007947, A055231.

KEYWORD

nonn,mult

AUTHOR

Henry Bottomley, Jun 18 2001

STATUS

approved

A073752

Greatest common divisor of n/spf(n) and n/gpf(n) where spf(n) is the smallest and gpf(n) is the greatest prime factor of n (see A020639, A006530).

+10
3

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 3, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 6, 1, 1, 3, 32, 1, 3, 1, 2, 1, 5, 1, 12, 1, 1, 5, 2, 1, 3, 1, 8, 27, 1, 1, 6, 1, 1, 1, 4, 1, 9, 1, 2, 1, 1, 1, 16, 1, 7, 3, 10, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

a(n) = if n=p^k (p prime, k>0) then p^(k-1) else n/(spf(n)*gpf(n)).

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = GCD(A032742(n), A052126(n)).

MATHEMATICA

gc[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]}, GCD[n/First[fi], n/Last[ fi]]]; Array[gc, 110] (* Harvey P. Dale, Jun 17 2012 *)

CROSSREFS

Cf. A000961, A066048.

A073753(n) = a(a(n)).

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Aug 07 2002

STATUS

approved

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