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A056623 Largest unitary square divisor of n: if n=LLgggf (see A056192) and a(n) = LL, then its complementary divisor n/LL = gggf and gcd(L^2, n/LL) = 1. 4
1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 1, 25, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 9, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 1, 49, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

EXAMPLE

n=250: largest square divisor is 25, but is not unitary. gcd(25,10) = 5 = A055229(250) = g, so a(250) = 25/g^2 = 1. The largest square and unitary divisor is here 1; n=200, A008833(200)=100, A055229(200) = g = gcd(100,2) = 2 so a(200) = 100/2^2 = 25 is already unitary because gcd(25, 200/25) = 1.

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

A056623(n) = (A008833(n)/(A055229(n)^2)); \\ Antti Karttunen, Nov 19 2017

(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

Cf. A008833, A055229, A000188, A046951, A034444, A056192, A056622.

Sequence in context: A046591 A119350 A016528 * A038025 A079982 A265143

Adjacent sequences:  A056620 A056621 A056622 * A056624 A056625 A056626

KEYWORD

nonn,mult,easy

AUTHOR

Labos Elemer, Aug 08 2000

STATUS

approved

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Last modified October 19 22:05 EDT 2021. Contains 348095 sequences. (Running on oeis4.)