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
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.
Crossrefs
Programs
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Mathematica
Table[With[{r = Apply[Times, FactorInteger[n][[All, 1]]]}, GCD[r, n/r]], {n, 104}] (* Michael De Vlieger, Aug 20 2017 *) -
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
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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
Formula
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
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)
Comments