|
|
A247513
|
|
Number of elements in the set {(x,y,z): 1<=x,y,z<=n, gcd(x,y,z)=1, lcm(x,y,z)=n}.
|
|
2
|
|
|
1, 6, 6, 12, 6, 36, 6, 18, 12, 36, 6, 72, 6, 36, 36, 24, 6, 72, 6, 72, 36, 36, 6, 108, 12, 36, 18, 72, 6, 216, 6, 30, 36, 36, 36, 144, 6, 36, 36, 108, 6, 216, 6, 72, 72, 36, 6, 144, 12, 72, 36, 72, 6, 108, 36, 108, 36, 36, 6, 432
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
For given n and k positive integers, let L(n,k) represent the number of ordered k-tuples of positive integers whose GCD is 1 and LCM is n. In this notation, the sequence corresponds to a(n) = L(n,3).
|
|
LINKS
|
|
|
FORMULA
|
For n = p_1^{n_1} p_2^{n_2}...p_r^{n_r} one has
a(n) = Product_{i=1..r} ((n_i+1)^3 - 2*n_i^3 + (n_i-1)^3).
a(n) = 6^omega(n)*Product_{i=1..r} n_i.
Multiplicative with a(p^e) = 6*e. - Amiram Eldar, Sep 26 2020
|
|
EXAMPLE
|
The triples corresponding to a(2)=6 are (1,1,2), (1,2,1), (2,1,1), (1,2,2), (2,1,2) and (2,2,1).
|
|
MAPLE
|
a:= proc(n) local F; F:= ifactors(n)[2];
mul(6*f[2], f=F)
end proc:
|
|
MATHEMATICA
|
a[n_] := 6^PrimeNu[n] Times @@ FactorInteger[n][[All, 2]];
a[1] = 1; a[n_] := Times @@ (6 * Last[#]& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)
|
|
PROG
|
(PARI) a(n) = {f = factor(n); 6^omega(n)*prod(k=1, #f~, f[k, 2]); }
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,mult,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|