A308474 a(n) = Sum_{k=1..n^2, gcd(n,k) = 1} k.

1, 4, 27, 64, 250, 216, 1029, 1024, 2187, 2000, 6655, 3456, 13182, 8232, 13500, 16384, 39304, 17496, 61731, 32000, 55566, 53240, 133837, 55296, 156250, 105456, 177147, 131712, 341446, 108000, 446865, 262144, 359370, 314432, 514500, 279936, 911754, 493848, 711828, 512000

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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.

a(n) = A000290(n)*A023896(n).

a(n) = A000578(n)*A023022(n) for n > 2.

Sum_{k=1..n} a(k) ~ 3*n^5/(5*Pi^2). - Vaclav Kotesovec, May 30 2019

Maple

f:= n -> n^3 * numtheory:-phi(n)/2:
f(1):= 1:
map(f, [$1..50]); # Robert Israel, Jan 22 2025

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

Cf. A000010, A000290, A000578, A002618, A008683, A023022, A023896.

Sequence in context: A078111 A186882 A097792 * A058067 A294038 A175701

Adjacent sequences: A308471 A308472 A308473 * A308475 A308476 A308477

Keyword

nonn

Author

Ilya Gutkovskiy, May 29 2019

Status

approved