A342432 -id:A342432 - cp's OEIS frontend

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.

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

A342433 a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).

Original entry on oeis.org

1, 3, 11, 74, 629, 8085, 117655, 2113796, 43059849, 1001955177, 25937424611, 743379914746, 23298085122493, 793811662313709, 29192938251553759, 1152956691126550536, 48661191875666868497, 2185928270773974154773

Offset: 1

Seiichi Manyama, Mar 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A000010, A000272, A321349, A332517, A332621, A342432, A343510.

a[n_] := Sum[GCD[k, n]^(n - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)

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

a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n-1));

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n-2));

a(n) = Sum_{d|n} phi(n/d) * d^(n-1).
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-2)(d).
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 23 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

A341036 a(n) = Sum_{k=1..n} gcd(k,n)^(k-2).

Original entry on oeis.org

1, 2, 5, 19, 129, 1306, 16813, 262181, 4783059, 100000214, 2357947701, 61917372083, 1792160394049, 56693912393474, 1946195068811453, 72057594039243049, 2862423051509815809, 121439531096661117354, 5480386857784802185957

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Cf. A000010, A018804, A342370, A342432, A342449.

a[n_] := Sum[GCD[k, n]^(k - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^(k-2));
```

cp's OEIS Frontend

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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A342433 a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).

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

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A341036 a(n) = Sum_{k=1..n} gcd(k,n)^(k-2).

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