cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A366362 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.

Original entry on oeis.org

1, 0, 4, 5, 0, 4, 0, 8, 0, 8, 21, 0, 0, 0, 4, 0, 20, 0, 0, 0, 16, 40, 0, 0, 0, 0, 0, 9, 0, 32, 0, 16, 0, 0, 0, 16, 45, 0, 24, 0, 0, 0, 0, 0, 12, 0, 84, 0, 0, 0, 0, 0, 0, 0, 16, 111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32
Offset: 1

Views

Author

Mats Granvik, Oct 08 2023

Keywords

Comments

Row n appears to have sum n^2. T(prime(m),1) = A366346(m). The number of nonzero terms in row n appears to be A320111(n).

Examples

			{
{1}, = 1^2
{0, 4}, = 2^2
{5, 0, 4}, = 3^2
{0, 8, 0, 8}, = 4^2
{21, 0, 0, 0, 4}, = 5^2
{0, 20, 0, 0, 0, 16}, = 6^2
{40, 0, 0, 0, 0, 0, 9}, = 7^2
{0, 32, 0, 16, 0, 0, 0, 16}, = 8^2
{45, 0, 24, 0, 0, 0, 0, 0, 12}, = 9^2
{0, 84, 0, 0, 0, 0, 0, 0, 0, 16}, = 10^2
{111, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 11^2
{0, 40, 0, 40, 0, 32, 0, 0, 0, 0, 0, 32} = 12^2
}

Crossrefs

Cf. A127649, A366346, A000290, A060457, A002070, A036689, A001248, A272196.

Programs

  • Mathematica
    f = x^3 - x^2 - y^2 - y; nn = 12; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^3 - x^2 - y^2 - y.
Conjecture: T(n,n) = A060457(n).

A366561 Triangle read by rows: T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^2 - y^2.

Original entry on oeis.org

1, 2, 2, 4, 0, 5, 8, 0, 0, 8, 16, 0, 0, 0, 9, 8, 8, 10, 0, 0, 10, 36, 0, 0, 0, 0, 0, 13, 32, 0, 0, 8, 0, 0, 0, 24, 36, 0, 24, 0, 0, 0, 0, 0, 21, 32, 32, 0, 0, 18, 0, 0, 0, 0, 18, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40
Offset: 1

Views

Author

Mats Granvik, Oct 13 2023

Keywords

Comments

Row n appears to have sum n^2. The number of nonzero terms in row n is A366563(n). Sum_{k=1..n} T(n,k)*A023900(k)/n = A366562(n).

Examples

			{
{1}, = 1^2
{2, 2}, = 2^2
{4, 0, 5}, = 3^2
{8, 0, 0, 8}, = 4^2
{16, 0, 0, 0, 9}, = 5^2
{8, 8, 10, 0, 0, 10}, = 6^2
{36, 0, 0, 0, 0, 0, 13}, = 7^2
{32, 0, 0, 8, 0, 0, 0, 24}, = 8^2
{36, 0, 24, 0, 0, 0, 0, 0, 21}, = 9^2
{32, 32, 0, 0, 18, 0, 0, 0, 0, 18}, = 10^2
{100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21}, = 11^2
{32, 0, 40, 32, 0, 0, 0, 0, 0, 0, 0, 40} = 12^2
}

Crossrefs

Cf. A127649, A000290, A062803, A082953.
Cf. A000010, A054523.

Programs

  • Mathematica
    nn = 12; f = x^2 - y^2; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[Sum[Sum[If[GCD[f, n] == k, 1, 0], {x, 1, n}], {y, 1, n}], {k, 1, n}], {n, 1, nn}]]
  • PARI
    T(n,k) = sum(x=1, n, sum(y=1, n, gcd(x^2 - y^2, n) == k)); \\ Michel Marcus, Oct 14 2023

Formula

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x^2 - y^2.
Conjecture 1: T(n,n) = A062803(n).
Conjecture 2: T(n,1) = A082953(n).
Conjectures from Ridouane Oudra, Jun 17 2025: (Start)
T(n,k) = 0 iff k not divide n.
T(n,k) = phi(n/k)*Sum_{d|k} (k/d)*phi(d*n/k), for n odd and k|n.
T(n,k) = 2*(-1)^k*phi(n/k)*Sum_{d|k} (-1)^(k/d)*(k/d)*phi(d*n/k), for n even and k|n.
T(n,k) = gcd(n,2)*(-1)^k*phi(n/k)*Sum_{d|k} (-1)^(k/d)*(k/d)*phi(d*n/k), for all integers n and k|n.
More generally, for all integers n, k: T(n,k) = gcd(n,2)*(-1)^k*A054523(n,k)*Sum_{d|k} (-1)^(k/d)*(k/d)*A054523(d*n,k). (End)

A366444 Triangle read by rows: T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, -1, 2, 0, -2, 2, -1, 0, -1, 4, 0, 0, 0, -4, 2, -2, -2, 0, 0, 2, 6, 0, 0, 0, 0, 0, -6, 4, -2, 0, -1, 0, 0, 0, -1, 6, 0, -4, 0, 0, 0, 0, 0, -2, 4, -4, 0, 0, -4, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Mats Granvik, Oct 12 2023

Keywords

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{1, -1}, = 0
{2, 0, -2}, = 0
{2, -1, 0, -1}, = 0
{4, 0, 0, 0, -4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{6, 0, 0, 0, 0, 0, -6}, = 0
{4, -2, 0, -1, 0, 0, 0, -1}, = 0
{6, 0, -4, 0, 0, 0, 0, 0, -2}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10}, = 0
{4, -2, -4, -2, 0, 2, 0, 0, 0, 0, 0, 2} = 0
}

Crossrefs

Cf. A000010, A023900, A366445, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

  • Mathematica
    nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, EulerPhi[n/k]*g[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = phi(n/k)*A023900(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

A366445 Triangle read by rows: T(n,k) = A023900(n/k)*phi(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, -1, 1, -2, 0, 2, -1, -1, 0, 2, -4, 0, 0, 0, 4, 2, -2, -2, 0, 0, 2, -6, 0, 0, 0, 0, 0, 6, -1, -1, 0, -2, 0, 0, 0, 4, -2, 0, -4, 0, 0, 0, 0, 0, 6, 4, -4, 0, 0, -4, 0, 0, 0, 0, 4, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 2, 2, -2, -4, 0, -2, 0, 0, 0, 0, 0, 4
Offset: 1

Views

Author

Mats Granvik, Oct 12 2023

Keywords

Comments

Sum_{k=1..n} T(n,k) = A063524(n).

Examples

			{
{1}, = 1
{-1, 1}, = 0
{-2, 0, 2}, = 0
{-1, -1, 0, 2}, = 0
{-4, 0, 0, 0, 4}, = 0
{2, -2, -2, 0, 0, 2}, = 0
{-6, 0, 0, 0, 0, 0, 6}, = 0
{-1, -1, 0, -2, 0, 0, 0, 4}, = 0
{-2, 0, -4, 0, 0, 0, 0, 0, 6}, = 0
{4, -4, 0, 0, -4, 0, 0, 0, 0, 4}, = 0
{-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10}, = 0
{2, 2, -2, -4, 0, -2, 0, 0, 0, 0, 0, 4} = 0
}

Crossrefs

Cf. A000010, A023900, A366444, A054524, A054525, A063524, A054522, A054523, A129691, A127649.

Programs

  • Mathematica
    nn = 12; g[n_] := DivisorSum[n, MoebiusMu[#] # &]; Flatten[Table[Table[If[Mod[n, k] == 0, g[n/k]*EulerPhi[k], 0], {k, 1, n}], {n, 1, nn}]]

Formula

T(n,k) = A023900(n/k)*phi(k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.