A368743 -id:A368743 - 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

A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

Original entry on oeis.org

1, 4, 9, 4, 25, 36, 49, 4, 9, 100, 121, 36, 169, 196, 225, 4, 289, 36, 361, 100, 441, 484, 529, 36, 25, 676, 9, 196, 841, 900, 961, 4, 1089, 1156, 1225, 36, 1369, 1444, 1521, 100, 1681, 1764, 1849, 484, 225, 2116, 2209, 36, 49, 100, 2601, 676, 2809, 36, 3025, 196

Offset: 1

Reinhard Zumkeller, Dec 10 2002

It is conjectured that only 1 and 1782 satisfy a(k) = sigma(k). See Broughan link. - Michel Marcus, Feb 28 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.
K. Broughan, J.-M. De Koninck, I. Kátai, and F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.
Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.

Cf. A002117, A007948, A055231, A062378, A330523, A007434, A008683, A001221, A065958.

a := n -> mul(f,f=map(x->x^2,select(isprime,divisors(n))));
seq(a(n), n=1..56);  # Peter Luschny, Mar 30 2014

a[n_] := Times @@ FactorInteger[n][[All, 1]]^2; Array[a, 60] (* Jean-François Alcover, Jun 04 2019 *)

a(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])^2 \\ Charles R Greathouse IV, Aug 06 2013

Multiplicative with a(p^e) = p^2. - Mitch Harris, May 17 2005
G.f.: Sum_{k>=1} mu(k)^2*J_2(k)*x^k/(1 - x^k), where J_2() is the Jordan function. - Ilya Gutkovskiy, Nov 06 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 / 3 = 0.214725... . - Amiram Eldar, Oct 30 2022
a(n) = Sum_{1 <= i, j <= n}  ( mobius(n/gcd(i, j, n)) )^2. - Peter Bala, Jan 28 2024
a(n) = Sum_{d|n} mu(d)^2*J_2(d), where J_2 = A007434. - Ridouane Oudra, Jul 24 2025
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*Psi_2(d), where omega = A001221 and Psi_2 = A065958. - Ridouane Oudra, Aug 01 2025

A372929 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^4.

Original entry on oeis.org

1, 23, 107, 424, 749, 2461, 2743, 7232, 9369, 17227, 15971, 45368, 30757, 63089, 80143, 119296, 88433, 215487, 137179, 317576, 293501, 367333, 292007, 773824, 483625, 707411, 777843, 1163032, 731669, 1843289, 953311, 1937408, 1708897, 2033959, 2054507, 3972456

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A343497, A368743, A372928, A372930.
Cf. A343498, A372926, A372931.
Cf. A000203, A008683.
Cf. A013661, A013663.

f[p_, e_] := p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^3*sigma(d));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^3.
a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-4)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)/zeta(5) = 1.586353589... . (End)

A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

Original entry on oeis.org

1, 19, 89, 316, 649, 1691, 2449, 5104, 7281, 12331, 14761, 28124, 28729, 46531, 57761, 81856, 83809, 138339, 130681, 205084, 217961, 280459, 280369, 454256, 406225, 545851, 590409, 773884, 708121, 1097459, 924481, 1310464, 1313729, 1592371, 1589401, 2300796

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A069097, A360428, A368743, A372927.
Cf. A343498, A372929, A372931.
Cf. A001157, A008683.
Cf. A002117, A013663.

f[p_, e_] := p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

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

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-4)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3)/zeta(5) = 1.1592484598... . (End)

A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

Original entry on oeis.org

1, 15, 53, 176, 249, 795, 685, 1856, 2133, 3735, 2661, 9328, 4393, 10275, 13197, 18432, 9825, 31995, 13717, 43824, 36305, 39915, 24333, 98368, 46625, 65895, 76545, 120560, 48777, 197955, 59581, 176128, 141033, 147375, 170565, 375408, 101305, 205755, 232829, 462144

Offset: 1

Seiichi Manyama, May 17 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Column k=3 of A372938.
Cf. A343497, A368743, A372929, A372930.
Cf. A000005, A008683.
Cf. A001620, A013662, A261506.

f[p_, e_] := (e - e/p^3 + 1) * p^(3*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^3*numdiv(d));

a(n) = Sum_{d|n} mu(n/d) * d^3 * tau(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (e - e/p^3 + 1) * p^(3*e).
Dirichlet g.f.: zeta(s-3)^2/zeta(s).
Sum_{k=1..n} a(k) ~ (n^4/(4*zeta(4))) * (log(n) + 2*gamma - 1/4 - zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)

A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

Original entry on oeis.org

1, 39, 269, 1304, 3249, 10491, 17149, 42176, 66069, 126711, 162381, 350776, 373489, 668811, 873981, 1353216, 1424769, 2576691, 2482957, 4236696, 4613081, 6332859, 6448509, 11345344, 10168625, 14566071, 16073721, 22362296, 20535537, 34085259, 28658941, 43331584

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A343497, A368743, A372928, A372929.
Cf. A001157, A008683.
Cf. A002117, A013664, A157289.

f[p_, e_] := p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^3*sigma(d, 2));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^3.
a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(3)/zeta(6) = 1.181564... (A157289). (End)

A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

Original entry on oeis.org

1, 35, 251, 1132, 3149, 8785, 16855, 36272, 61065, 110215, 161171, 284132, 371461, 589925, 790399, 1160896, 1420145, 2137275, 2476459, 3564668, 4230605, 5640985, 6436871, 9104272, 9841225, 13001135, 14839443, 19079860, 20511989, 27663965, 28630111, 37149440, 40453921

Offset: 1

Seiichi Manyama, May 17 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A069097, A360428, A368743, A372926.
Cf. A001158, A008683.
Cf. A013662, A013664, A088246.

f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 3));

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End)

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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A078615 a(n) = rad(n)^2, where rad is the squarefree kernel of n (A007947).

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A372929 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^4.

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A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

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A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

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A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

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A372927 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

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