A332685 -id:A332685 - cp's OEIS frontend

A332686 a(n) = Sum_{k=1..n} phi(k/gcd(n, k)).

1, 2, 3, 5, 7, 9, 13, 18, 21, 23, 33, 33, 47, 49, 51, 67, 81, 76, 103, 97, 103, 119, 151, 135, 163, 173, 185, 189, 243, 185, 279, 280, 265, 299, 291, 291, 397, 379, 369, 371, 491, 381, 543, 491, 455, 553, 651, 539, 653, 610, 643, 683, 831, 689, 743, 753, 801, 887, 1029

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Inverse Moebius transform of A053570.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A002088, A029935, A029939, A053570, A332685.

Table[Sum[EulerPhi[k/GCD[n, k]], {k, 1, n}], {n, 1, 59}]

a(n) = sum(k=1, n, eulerphi(k/gcd(n, k))); \\ Michel Marcus, Feb 21 2020

a(n) = Sum_{k=1..n} phi(lcm(n, k)/n).

a(n) = Sum_{d|n} A053570(d).

A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A001694 (positions of nonzero terms), A005361, A007427, A008683, A008836, A028242, A052485 (positions of 0's), A062838 (positions of 1's), A112526, A252505, A322483, A332685, A332713.

Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]
A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
f[p_, e_] := 3*Floor[e/2] - e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(2*s)^2 * zeta(3*s) / zeta(6*s).
a(n) = Sum_{d|n} mu(lcm(d, n/d)/d).
a(n) = Sum_{d|n} (-1)^bigomega(n/d) * A005361(d).
a(n) = Sum_{d|n} A010052(n/d) * A112526(d).
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)*log(n)/(2*zeta(3)) + ((2*gamma - 1)*zeta(3/2) + 3*zeta'(3/2)/2 - 3*zeta(3/2)*zeta'(3)/zeta(3)) * sqrt(n)/zeta(3) + 6*zeta(2/3)^2 * n^(1/3)/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 21 2020
Multiplicative with a(p^e) = A028242(e). - Amiram Eldar, Nov 30 2020

cp's OEIS Frontend

A332686 a(n) = Sum_{k=1..n} phi(k/gcd(n, k)).

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