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.
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, 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, 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
Offset: 1
Examples
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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..19683
Programs
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Mathematica
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 *) -
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
Formula
a(n) = A055229(n)^3 = g^3 = ggg; n = (KK)*(ggg)*f = K^2*g^3*f = KK*a(n)^3*f.
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
Comments