A071773 - OEIS

%I #44 Nov 09 2023 08:54:31

%S 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,

%T 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,

%U 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

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

%C n is squarefree iff a(n)=1.

%C Product of primes dividing n more than once. - _Charles R Greathouse IV_, Aug 08 2013

%C Squarefree kernel of the square part of n. - _Peter Munn_, Jun 12 2020

%H Charles R Greathouse IV, <a href="/A071773/b071773.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.

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

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

%F a(n) = rad(n/rad(n)) = A007947(A003557(n)). - _Velin Yanev_, _Antti Karttunen_, Aug 20 2017, Nov 28 2017

%F a(n) = A007947(A057521(n)). - _Antti Karttunen_, Nov 28 2017

%F a(n) = A007947(A008833(n)). - _Peter Munn_, Jun 12 2020

%F a(n) = gcd(A003415(n), A007947(n)). - _Antti Karttunen_, Jan 02 2023

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

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

%o (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

%o (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

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

%K nonn,easy,mult

%O 1,4

%A _Reinhard Zumkeller_, Jun 24 2002