A029939 -id:A029939 - cp's OEIS frontend

A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

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

Seiichi Manyama, Mar 15 2021

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

Cf. A000010, A029935, A029939, A065464, A306633, A338997, A342535.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^2 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)

a(n) = sum(k=1, n, eulerphi(gcd(k, n))^2);

a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^2);

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)

A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

Original entry on oeis.org

1, 2, 1, 5, 0, 2, 6, 3, 0, 2, 17, 0, 0, 0, 4, 10, 5, 4, 0, 0, 2, 37, 0, 0, 0, 0, 0, 6, 22, 11, 0, 6, 0, 0, 0, 4, 41, 0, 14, 0, 0, 0, 0, 0, 6, 34, 17, 0, 0, 8, 0, 0, 0, 0, 4, 101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 30, 15, 12, 10, 0, 6, 0, 0, 0, 0, 0, 4, 145, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 74, 37, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054522 and A054523 are commuted, the matrix product becomes A127478.

			First few rows of the triangle are:
1;
2, 1;
5, 0, 2;
6, 3, 0, 2;
17, 0, 0, 0, 4;
10, 5, 4, 0, 0, 2;
37, 0, 0, 0, 0, 0, 6;
22, 11, 0, 6, 0, 0, 0, 4;

Cf. A054522, A054523, A057660, A000010, A029939.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127477 := proc(n,k) add( A054522(n,j)*A054523(j,k), j=k..n) ; end: seq(seq( A127477(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054522(n,j) * A054523(j,k).
sum_{k=1..n} T(n,k) = A057660(n) (row sums).
T(n,n) = A000010(n) (diagonal).
T(n,1) = A029939(n).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

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

Original entry on oeis.org

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).

A127479 Triangle read by rows: A054522 * A054521 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 0, 2, 0, 5, 4, 4, 4, 0, 6, 2, 0, 0, 2, 0, 7, 6, 6, 6, 6, 6, 0, 8, 0, 6, 0, 4, 0, 4, 0, 9, 8, 0, 6, 6, 0, 6, 6, 0, 10, 4, 8, 4, 0, 0, 4, 0, 4, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

Row sums = A029939: (1, 2, 5, 6, 17, 10, 37, ...).

			First few rows of the triangle:
  1;
  2, 0;
  3, 2, 0;
  4, 0, 2, 0;
  5, 4, 4, 4, 0;
  6, 2, 0, 0, 2, 0;
  7, 6, 6, 6, 6, 6, 0;
  8, 0, 6, 0, 4, 0, 4, 0;
  ...

Cf. A029939, A054522, A054521.

Edited by N. J. A. Sloane, Aug 10 2019

cp's OEIS Frontend

A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

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A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

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

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Mathematica

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Formula

A127479 Triangle read by rows: A054522 * A054521 as infinite lower triangular matrices.

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