A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 10, 1, 1, 1, 2

Offset

1,4

Comments

n is squarefree iff a(n)=1.

Product of primes dividing n more than once. - Charles R Greathouse IV, Aug 08 2013

Squarefree kernel of the square part of n. - Peter Munn, Jun 12 2020

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Eric Weisstein's World of Mathematics, Squarefree.

Formula

a(n) = gcd(A007947(n), A003557(n)).

Multiplicative with p^e -> p^ceiling((e-1)/e), p prime.

a(n) = rad(n/rad(n)) = A007947(A003557(n)). - Velin Yanev, Antti Karttunen, Aug 20 2017, Nov 28 2017

a(n) = A007947(A057521(n)). - Antti Karttunen, Nov 28 2017

a(n) = A007947(A008833(n)). - Peter Munn, Jun 12 2020

a(n) = gcd(A003415(n), A007947(n)). - Antti Karttunen, Jan 02 2023

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Nov 09 2023

From Vaclav Kotesovec, May 06 2025: (Start)

Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(4*s-1) - 1/p^(4*s-2)).

Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where

f(1) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.428249505677094440218765707581823546121298513355936144031901379532123...

f'(1) = f(1) * Sum_{p prime} 2*(3*p-2)*log(p) / (p^3-2*p+1) = f(1) * 2.939073481649229666406787986900328729326669597518287791424059647447664...

and gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

Table[With[{r = Apply[Times, FactorInteger[n][[All, 1]]]}, GCD[r, n/r]], {n, 104}] (* Michael De Vlieger, Aug 20 2017 *)

Prog

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]>1)) \\ Charles R Greathouse IV, Aug 08 2013
(Scheme)
;; With memoization-macro definec.
(definec (A071773 n) (if (= 1 n) n (* (if (zero? (modulo n (expt (A020639 n) 2))) (A020639 n) 1) (A071773 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Crossrefs

Cf. A003415, A003557, A005117, A007947, A007948, A008833, A057521, A166486 (parity of terms), A359433 (Dirichlet inverse).

Cf. A065464.

Sequence in context: A370784 A249739 A249740 * A308993 A000188 A373833

Adjacent sequences: A071770 A071771 A071772 * A071774 A071775 A071776

Keyword

nonn,easy,mult

Author

Reinhard Zumkeller, Jun 24 2002

Status

approved