A373059 -id:A373059 - cp's OEIS frontend

A371492 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^2.

1, 17, 91, 289, 701, 1547, 2647, 4769, 7705, 11917, 15731, 26299, 30421, 44999, 63791, 77473, 87857, 130985, 136459, 202589, 240877, 267427, 290951, 433979, 448201, 517157, 633187, 764983, 729989, 1084447, 951391, 1248929, 1431521, 1493569, 1855547, 2226745

Offset: 1

Seiichi Manyama, May 24 2024

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

Cf. A084218, A373060.
Cf. A372952, A372962.
Cf. A373059, A371628.
Cf. A002117, A013661, A013663.

f[p_, e_] := (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 36] (* Amiram Eldar, May 24 2024 *)

a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^2*sigma(d^2, 4)/sigma(d^2, 2));

a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_4(d^2)/sigma_2(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)).
Dirichlet g.f.: zeta(s)*zeta(s-3)*zeta(s-4)/zeta(s-2)^2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)*zeta(5)/zeta(3)^2 = 1.180448217... . (End)

A371628 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^3.

Original entry on oeis.org

1, 65, 757, 4225, 16001, 49205, 119365, 271489, 554797, 1040065, 1783541, 3198325, 4850977, 7758725, 12112757, 17392769, 24211265, 36061805, 47162485, 67604225, 90359305, 115930165, 148291397, 205517173, 250266001, 315313505, 404686153, 504317125, 595481825

Offset: 1

Seiichi Manyama, May 24 2024

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

Cf. A373059, A371492.
Cf. A372963, A372964.
Cf. A084220.
Cf. A002117, A013662, A013665.

f[p_, e_] := (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, May 24 2024 *)

a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 6)/sigma(d^2, 3));

a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_6(d^2)/sigma_3(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)).
Dirichlet g.f.: zeta(s)*zeta(s-4)*zeta(s-6)/zeta(s-3)^2.
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(3)*zeta(7)/zeta(4)^2 = 1.034718122... . (End)

A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 8, 5, 8, 4, 9, 9, 9, 9, 5, 15, 10, 9, 10, 15, 6, 13, 13, 13, 13, 13, 13, 7, 20, 12, 20, 9, 20, 12, 20, 8, 21, 21, 11, 21, 21, 11, 21, 21, 9, 27, 18, 27, 18, 15, 18, 27, 18, 27, 10, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 11, 40, 25, 24, 20, 40, 15, 40, 20, 24, 25, 40, 12

Offset: 1

Werner Schulte, Jan 21 2022

Subtriangle (triangle without main diagonal) is symmetrical.

Conjecture: Let f be an arbitrary arithmetic function. Define for n > 0 the sequence a(f; n) = Sum_{i=1..n, k=1..n} f(gcd(i,n)/gcd(gcd(i,k),n)); a(f; n) equals Dirichlet convolution of f(n)*A000010(n) and A057660(n); if f is multiplicative, then a(f; n) is multiplicative; row sums of this triangle use f(n) = n (see formula section).

			The triangle T(n, k) for 1 <= k <= n starts:
n \k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   3   2
   3 :   5   5   3
   4 :   8   5   8   4
   5 :   9   9   9   9   5
   6 :  15  10   9  10  15   6
   7 :  13  13  13  13  13  13   7
   8 :  20  12  20   9  20  12  20   8
   9 :  21  21  11  21  21  11  21  21   9
  10 :  27  18  27  18  15  18  27  18  27  10
  11 :  21  21  21  21  21  21  21  21  21  21  11
  12 :  40  25  24  20  40  15  40  20  24  25  40  12
  etc.

Row sums gives A373059.

Cf. A000010, A002618, A018804, A057660, A050873.

T(n, k) = sum(i=1, n, gcd(i,n) / gcd(gcd(i,k),n));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Jan 22 2022

T(n, 1) = A018804(n); T(n, n) = n.
T(n, k) = T(n, n-k) for 1 <= k < n.
Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.

cp's OEIS Frontend

A371492 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^2.

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A371628 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^3.

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A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

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