login
A342534
a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.
3
1, 2, 6, 7, 20, 12, 42, 26, 50, 40, 110, 42, 156, 84, 120, 100, 272, 100, 342, 140, 252, 220, 506, 156, 484, 312, 438, 294, 812, 240, 930, 392, 660, 544, 840, 350, 1332, 684, 936, 520, 1640, 504, 1806, 770, 1000, 1012, 2162, 600, 2022, 968, 1632, 1092, 2756, 876, 2200, 1092, 2052
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{d|n} phi(n/d) * phi(d)^2.
a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n)). - Richard L. Ollerton, May 10 2021
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = (p-1)*(p^(e-2) - p^(2*e-3) + p^(2*e-1)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)/(3*zeta(3)) * Product_{p prime} (1 - (2*p-1)/p^3) = A306633 * A065464 / 3 = 0.195343... . (End)
MATHEMATICA
a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^2 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)
PROG
(PARI) a(n) = sum(k=1, n, eulerphi(gcd(k, n))^2);
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^2);
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, Mar 15 2021
STATUS
approved