OFFSET
1,2
FORMULA
G.f.: Sum_{k>=1} mu(k)*k^3*x^k*(1 + 7*x^k + 4*x^(2*k))/(1 - x^k)^5.
a(n) = n^3*phi(n)/2 for n > 1.
a(n) = n^3 * Sum_{d|n} mu(n/d)*(d + 1)/2.
Sum_{k=1..n} a(k) ~ 3*n^5/(5*Pi^2). - Vaclav Kotesovec, May 30 2019
MATHEMATICA
a[n_] := Sum[If[GCD[n, k] == 1, k, 0], {k, 1, n^2}]; Table[a[n], {n, 1, 40}]
nmax = 40; CoefficientList[Series[Sum[MoebiusMu[k] k^3 x^k (1 + 7 x^k + 4 x^(2 k))/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Join[{1}, Table[n^3 EulerPhi[n]/2, {n, 2, 40}]]
Table[n^3 Sum[MoebiusMu[n/d] (d + 1)/2, {d, Divisors[n]}], {n, 1, 40}]
PROG
(PARI) a(n) = sum(k=1, n^2, if (gcd(n, k)==1, k)); \\ Michel Marcus, May 31 2019
(Python)
from sympy import totient
def A308474(n): return n**3*totient(n)>>1 if n>1 else 1 # Chai Wah Wu, Nov 06 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 29 2019
STATUS
approved