A130586 -id:A130586 - cp's OEIS frontend

A332621 a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).

1, 3, 19, 133, 2501, 15631, 705895, 8389641, 258280489, 4000040011, 259374246011, 2972033984173, 279577021469773, 4762288684702095, 233543408203327951, 9223372037928525841, 778579070010669895697, 13115469358498302735067, 1874292305362402347591139

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A000010, A056665, A130586, A226561, A228640, A308814, A321349, A332620.

[(1/n)*&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020

Table[(1/n) Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[(1/n) Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^(j - 1) x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]

a(n) = sum(k=1, n, n^(n/gcd(n, k)))/n; \\ Michel Marcus, Mar 10 2021

a(n) = [x^n] Sum_{k>=1} Sum_{j>=1} phi(j) * n^(j-1) * x^(k*j).
a(n) = (1/n) * Sum_{k=1..n} n^(lcm(n, k)/k).
a(n) = (1/n) * Sum_{d|n} phi(d) * n^d.
a(n) = A332620(n) / n.

A130585 A054522 * A007318.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 7, 6, 2, 5, 16, 24, 16, 4, 6, 15, 22, 20, 10, 2, 7, 36, 90, 120, 90, 36, 6, 8, 35, 90, 142, 140, 84, 28, 4, 9, 52, 170, 336, 420, 336, 168, 48, 6, 10, 53, 168, 352, 508, 504, 336, 144, 36, 4

Offset: 0

Gary W. Adamson, Jun 06 2007

A130584 = A007318 * A054522 Row sums = A130586: (1, 3, 9, 19, 65, 75, 385, ...).

			First few rows of the triangle:
  1;
  2,  1;
  3,  4,  2;
  4,  7,  6,   2;
  5, 16, 24,  16,  4;
  6, 15, 22,  20, 10,  2;
  7, 36, 90, 120, 90, 36, 6;
  ...

Cf. A054522, A007318, A130585, A130586.

A054522 * A007318 as infinite lower triangular matrices.

A338647 a(n) = Sum_{k=1..n} 2^(k/gcd(n,k) - 1).

Original entry on oeis.org

1, 2, 4, 7, 16, 22, 64, 92, 223, 342, 1024, 1132, 4096, 5462, 13534, 21937, 65536, 70978, 262144, 333472, 890590, 1398102, 4194304, 4528402, 16236031, 22369622, 57522106, 88435312, 268435456, 272976502, 1073741824, 1431677702, 3679303390, 5726623062, 16490405374, 18543422953

Offset: 1

Ilya Gutkovskiy, Apr 22 2021

nonn

Cf. A011782, A034738, A054432, A057661, A130586.

Table[Sum[2^(k/GCD[n, k] - 1), {k, 1, n}], {n, 1, 36}]

a(n) = sum(k=1, n, 2^(k/gcd(n,k) - 1)); \\ Michel Marcus, Apr 22 2021

a(n) = Sum_{d|n} A054432(d).

cp's OEIS Frontend

A332621 a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).

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A130585 A054522 * A007318.

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A338647 a(n) = Sum_{k=1..n} 2^(k/gcd(n,k) - 1).

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