A321294 -id:A321294 - cp's OEIS frontend

A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

1, 5, 29, 274, 3129, 47515, 823549, 16843268, 387459861, 10009769725, 285311670621, 8918311856102, 302875106592265, 11112685048729175, 437893951473411261, 18447025557276459016, 827240261886336764193, 39346558373052524325225, 1978419655660313589123997

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

If n is prime, a(n) = n-1 + n^n. - Robert Israel, Feb 16 2020

Robert Israel, Table of n, a(n) for n = 1..386

Cf. A000010, A008683, A018804, A031971, A069097, A226561, A228640, A321294.

[&+[Gcd(n,k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020

f:= n -> add(igcd(n,k)^n,k=1..n):
map(f, [$1..30]); # Robert Israel, Feb 16 2020

Table[Sum[GCD[n, k]^n, {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[n/d] d^n, {d, Divisors[n]}], {n, 1, 19}]
Table[Sum[MoebiusMu[n/d] d DivisorSigma[n - 1, d], {d, Divisors[n]}], {n, 1, 19}]

a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 2020

from sympy import totient, divisors
def A332517(n):
    return sum(totient(d)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = Sum_{d|n} phi(n/d) * d^n.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-1)(d).
a(n) ~ n^n.
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^n*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k)*sigma_(n-1)(gcd(n,k))/phi(n/gcd(n,k)). (End)

A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 9, 1, 33, 83, 74, 29, 15, 1, 65, 245, 274, 129, 55, 13, 1, 129, 731, 1058, 629, 261, 55, 20, 1, 257, 2189, 4162, 3129, 1411, 349, 92, 21, 1, 513, 6563, 16514, 15629, 8085, 2407, 596, 105, 27, 1, 1025, 19685, 65794, 78129, 47515, 16813, 4388, 789, 145, 21

Offset: 1

Seiichi Manyama, Apr 17 2021

			G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
Square array begins:
   1,  1,   1,    1,     1,      1,      1, ...
   3,  5,   9,   17,    33,     65,    129, ...
   5, 11,  29,   83,   245,    731,   2189, ...
   8, 22,  74,  274,  1058,   4162,  16514, ...
   9, 29, 129,  629,  3129,  15629,  78129, ...
  15, 55, 261, 1411,  8085,  47515, 282381, ...
  13, 55, 349, 2407, 16813, 117655, 823549, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.
Eric Weisstein's World of Mathematics, Eulerian Number and Euler's Number Triangle

Columns k=1..7 give A018804, A069097, A343497, A343498, A343499, A343508, A343509.
T(n-2,n) gives A342432.
T(n-1,n) gives A342433.
T(n,n) gives A332517.
T(n,n+1) gives A321294.
Cf. A008292, A343516.

T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * #^k &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)

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

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

T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));

G.f. of column k: Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^(k+1).
T(n,k) = Sum_{d|n} phi(n/d) * d^k.
T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).
Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
T(n,k) = Sum_{j=1..n} (n/gcd(n,j))^k*phi(gcd(n,j))/phi(n/gcd(n,j)). - Richard L. Ollerton, May 10 2021
T(n,k) = Sum_{1 <= j_1, j_2, ..., j_k <= n} gcd(j_1, j_2, ..., j_k)^2 = Sum_{d divides n} d * J_k(n/d), where J_k(n) denotes the k-th Jordan totient function. - Peter Bala, Jan 29 2024

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A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

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A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

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