A332517 -id:A332517 - cp's OEIS frontend

A342424 a(n) = Sum_{k=1..n} gcd(k,n)^(n/gcd(k,n)).

1, 3, 5, 10, 9, 33, 13, 60, 69, 167, 21, 470, 25, 837, 1245, 1624, 33, 5067, 37, 9570, 13841, 20633, 45, 54612, 12545, 98511, 119601, 201646, 57, 562957, 61, 794928, 1774185, 2097491, 536037, 5381754, 73, 9437601, 19136333, 14296940

Offset: 1

Seiichi Manyama, Mar 11 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000010, A332517, A342423.

a[n_] := Sum[GCD[k, n]^(n/GCD[k, n]), {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Mar 11 2021 *)

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

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

a(n) = Sum_{d|n} phi(n/d) * d^(n/d).

A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

Original entry on oeis.org

1, 5, 29, 262, 3129, 46705, 823549, 16777544, 387421251, 10000003469, 285311670621, 8916100581446, 302875106592265, 11112006826387025, 437893890391180013, 18446744073743123788, 827240261886336764193, 39346408075299116257065

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Cf. A000010, A001923, A018804, A050873, A231712, A332517, A341036, A342370, A342389, A342423.

a[n_] := Sum[GCD[k, n]^k, {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^k);
```

If p is prime, a(p) = p-1 + p^p = A231712(p).

A342539 a(n) = Sum_{k=1..n} phi(gcd(k, n))^n.

Original entry on oeis.org

1, 2, 10, 19, 1028, 132, 279942, 65798, 10078726, 2097160, 100000000010, 16797702, 106993205379084, 156728328204, 35186519703560, 281479271809036, 295147905179352825872, 203119914385420, 708235345355337676357650, 1152924803145924620, 46005163783270994804748, 20000000000000000000020

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000010, A029935, A332517, A342487, A342534, A342535, A342540, A342541, A342543.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)

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

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

a(n) = Sum_{d|n} phi(n/d) * phi(d)^n.
If p is prime, a(p) = p-1 + (p-1)^p.
a(n) = Sum_{k=1..n} phi(n/gcd(n,k))^(n-1)*phi(gcd(n,k)). - Richard L. Ollerton, May 09 2021

A321294 a(n) = Sum_{d|n} mu(n/d)dsigma_n(d).

Original entry on oeis.org

1, 9, 83, 1058, 15629, 282381, 5764807, 134480900, 3486902505, 100048836321, 3138428376731, 107006403495850, 3937376385699301, 155572843119518781, 6568408661060858767, 295150157013526773768, 14063084452067724991025, 708236697425777157039381

Offset: 1

Ilya Gutkovskiy, Nov 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A018804, A069097, A320940, A332517, A342432, A342433.

Table[Sum[MoebiusMu[n/d] d DivisorSigma[n, d], {d, Divisors[n]}], {n, 18}]
Table[Sum[EulerPhi[n/d] d^(n + 1), {d, Divisors[n]}], {n, 18}]
Table[Sum[GCD[n, k]^(n + 1), {k, n}], {n, 18}]

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n)); \\ Michel Marcus, Nov 03 2018

from sympy import totient, divisors
def A321294(n):
    return sum(totient(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = [x^n] Sum_{i>=1} Sum_{j>=1} mu(i)*j^(n+1)*x^(i*j)/(1 - x^(i*j))^2.
a(n) = Sum_{d|n} phi(n/d)*d^(n+1).
a(n) = Sum_{k=1..n} gcd(n,k)^(n+1).
a(n) ~ n^(n+1). - Vaclav Kotesovec, Nov 02 2018

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

Original entry on oeis.org

1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A031971, A057661, A226561, A332517, A332621, A332652, A332653, A332654.

[&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

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

a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} k^n.

cp's OEIS Frontend

A342424 a(n) = Sum_{k=1..n} gcd(k,n)^(n/gcd(k,n)).

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A342449 a(n) = Sum_{k=1..n} gcd(k,n)^k.

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A342539 a(n) = Sum_{k=1..n} phi(gcd(k, n))^n.

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A321294 a(n) = Sum_{d|n} mu(n/d)*d*sigma_n(d).

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A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

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A321294 a(n) = Sum_{d|n} mu(n/d)dsigma_n(d).