cp's OEIS Frontend

A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(K*K) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2*L^2 and g divides F = gf.

%I A056191 #20 Sep 05 2023 01:41:33
%S A056191 1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,1,1,27,1,1,1,1,8,1,1,
%T A056191 1,1,1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,27,1,8,1,1,1,1,1,1,1,1,1,1,1,1,
%U A056191 1,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,1,1,1,1,1,1,1,8,1,1,1,1,1,1,1
%N A056191 Characteristic cube divisor of n: cube of g = gcd(K,F), where K is the largest square root divisor of n (A000188) and F = n/(K*K) = A007913(n) is its squarefree part; g^2 divides K^2 = A008833(n) = g^2*L^2 and g divides F = gf.
%C A056191 This is not the largest cube which divides n. It is canonical, since the decomposition n = KKgggf is unique (factors are defined above and dependent on n).
%H A056191 Antti Karttunen, <a href="/A056191/b056191.txt">Table of n, a(n) for n = 1..19683</a>
%F A056191 a(n) = A055229(n)^3 = g^3 = ggg; n = (KK)*(ggg)*f = K^2*g^3*f = KK*a(n)^3*f.
%F A056191 Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p^3 for odd e > 1. - _Vladeta Jovovic_, May 01 2002
%e A056191 If n=24, largest square divisor is 4, squarefree part is 6, g=2, a(24)=8; n=81, largest square divisor is 81, both F and g is 1, a(81)=1.
%t A056191 a[n_]:=With[{sf=Times@@Power@@@({#[[1]], Mod[#[[2]], 2]}&/@FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n]^3, {n, 1, 100}] (* _Vincenzo Librandi_, Oct 08 2017 *)
%o A056191 (Scheme, with memoization-macro definec, after _Vladeta Jovovic_'s multiplicative formula)
%o A056191 (definec (A056191 n) (cond ((= 1 n) n) ((or (= 1 (A067029 n)) (even? (A067029 n))) (A056191 (A028234 n))) (else (* (A000578 (A020639 n)) (A056191 (A028234 n)))))) ;; _Antti Karttunen_, Oct 07 2017
%o A056191 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1 || !(f[i,2]%2), 1,  f[i,1]^3));} \\ _Amiram Eldar_, Sep 05 2023
%Y A056191 Cf. A055229, A000188, A008833, A007913, A055231, A056192.
%K A056191 nonn,mult
%O A056191 1,8
%A A056191 _Labos Elemer_, Aug 02 2000