OFFSET
1,4
COMMENTS
The part of the name "Largest unitary square divisor of n" was removed because it is correct only for numbers whose odd exponents in their prime factorization are all smaller than 5. For the correct largest unitary square divisor of n see A350388. - Amiram Eldar, Jul 26 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = A008833(n)/A055229(n)^2 = K^2/g^2, which coincides with the largest square divisor iff the g-factor is 1.
Multiplicative with a(p^e)=p^e for even e, a(p)=1, a(p^e)=p^(e-3) for odd e > 1. - Vladeta Jovovic, Apr 30 2002
From Amiram Eldar, Dec 25 2023 (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2) + 1/p^(9/2)) = 1.81133051934001073532... . (End)
a(n) = A056622(n)^2. - Amiram Eldar, Jul 26 2024
EXAMPLE
MATHEMATICA
f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^e, p^(e-3)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)
PROG
(PARI)
A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ Charles R Greathouse IV, Nov 20 2012
A008833(n) = n/core(n) \\ Michael B. Porter, Oct 17 2009
(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e == 1, 1, if(e%2, p^(e-3), p^e))); } \\ Amiram Eldar, Dec 25 2023
(Scheme) (definec (A056623 n) (if (= 1 n) n (let ((e (A067029 n)) (rest (A056623 (A028234 n)))) (cond ((even? e) (* (A028233 n) rest)) ((= 1 e) rest) (else (* (expt (A020639 n) (- e 3)) rest)))))) ;; After Jovovic's multiplicative formula, using memoization-macro definec - Antti Karttunen, Nov 19 2017
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Labos Elemer, Aug 08 2000
EXTENSIONS
Name edited by Amiram Eldar, Jul 26 2024
STATUS
approved