A054523 -id:A054523 - cp's OEIS frontend

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.

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

Mats Granvik, Oct 13 2023

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).

			{
{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
}

Cf. A127649, A000290, A062803, A082953.

Cf. A000010, A054523.

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}]]

T(n,k) = sum(x=1, n, sum(y=1, n, gcd(x^2 - y^2, n) == k)); \\ Michel Marcus, Oct 14 2023

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)

A127373 Triangle, row sums = A023896, left column = A053570.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 0, 1, 0, 6, 2, 1, 1, 0, 5, 0, 0, 0, 1, 0, 12, 4, 2, 1, 1, 1, 0, 13, 0, 1, 0, 1, 0, 1, 0, 18, 4, 0, 2, 1, 0, 1, 1, 0, 15, 0, 3, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Gary W. Adamson, Jan 12 2007

Row sums = A023896: (1, 1, 3, 4, 10, 6, ...); left column = A053570: (1, 2, 3, 6, 5, 12, ...).

			First few rows of the triangle:
   1;
   1, 0;
   2, 1, 0;
   3, 0, 1, 0;
   6, 2, 1, 1, 0;
   5, 0, 0, 0, 1, 0
  12, 4, 2, 1, 1, 1, 0;
  ...

Cf. A054521, A054523, A053570, A023896.

A054521 * A054523 as infinite lower triangular matrices.

A127465 Triangle read by rows: T(n,k) = k*phi(n/k) if k|n, T(n,k)=0 otherwise.

Original entry on oeis.org

1, 1, 2, 2, 0, 3, 2, 2, 0, 4, 4, 0, 0, 0, 5, 2, 4, 3, 0, 0, 6, 6, 0, 0, 0, 0, 0, 7, 4, 4, 0, 4, 0, 0, 0, 8, 6, 0, 6, 0, 0, 0, 0, 0, 9, 4, 8, 0, 0, 5, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 4, 4, 6, 8, 0, 6, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson, Jan 15 2007

			First few rows of the triangle are:
1;
1, 2;
2, 0, 3;
2, 2, 0, 4;
4, 0, 0, 0, 5;
2, 4, 3, 0, 0, 6;
6, 0, 0, 0, 0, 0, 7;
4, 4, 0, 4, 0, 0, 0, 8;
...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A054523, A000010 (column k=1), A018804 (row sums).

A127465 := proc(n,k)
        if n mod k = 0 then
                k*numtheory[phi](n/k) ;
        else
                0;
        end if;
end proc:
seq(seq(A127465(n,m),m=1..n),n=1..12) ; # R. J. Mathar, Nov 08 2011

T(n,k) = k*A054523(n,k).

A130030 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * prime(d).

Original entry on oeis.org

2, 1, 1, 2, 3, 6, 5, 7, 9, 14, 11, 15, 17, 20, 21, 22, 27, 20, 31, 23, 33, 38, 39, 20, 45, 48, 43, 35, 53, 6, 67, 47, 65, 64, 63, 25, 85, 78, 73, 34, 99, 20, 107, 63, 45, 94, 119, 35, 113, 56, 99, 73, 137, 54, 103, 54, 117, 134, 161, -1, 163, 136, 73, 96, 113, 24, 199, 107, 159, 12, 213

Offset: 1

Gary W. Adamson, May 02 2007

sign

Old name: A129691 * A000040.

A130029 = A054523 * A000040.

			a(4) = 2 = dot product of row 4 of A129691: (-1, -1, 0, 1) and the first four primes: (2, 3, 5, 7) = (-2, -3, 0, 7) = 2.

Cf. A000010, A000040, A001221, A007444, A007947, A023900, A054523, A076479, A129691, A130029, A333177.

A129691 as an infinite lower triangular matrix * A000040, the primes.
From Ilya Gutkovskiy, Mar 22 2020: (Start)
a(n) = Sum_{d|n} A023900(n/d) * prime(d).
Sum_{k=1..n} a(gcd(n,k)) = prime(n). (End)

New name and more terms from Ilya Gutkovskiy, Mar 22 2020

A228094 Triangle starting at row 3 read by rows of the number of permutations in the n-th Dihedral group which are the product of k disjoint cycles, d(n,k), n >= 3, 1 <= k <= n.

Original entry on oeis.org

2, 3, 1, 2, 3, 2, 1, 4, 0, 5, 0, 1, 2, 2, 4, 3, 0, 1, 6, 0, 0, 7, 0, 0, 1, 4, 2, 0, 5, 4, 0, 0, 1, 6, 0, 2, 0, 9, 0, 0, 0, 1, 4, 4, 0, 0, 6, 5, 0, 0, 0, 1, 10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 7, 6, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 1

Offset: 3

Robert A. Beeler, Aug 09 2013

The multivariable row polynomials give n times the cycle index for the Dihedral group D_n, called Z(D_n) (see the MathWorld link with the Harary reference). For example, 12*Z(D_6) = 2*(y_6)^1 + 2*(y_3)^2 + 4*(y_2)^3+3*(y_1)^2*(y_2)^2 + 1*(y_1)^6.

			Triangle begins
   2, 3, 1;
   2, 3, 2, 1;
   4, 0, 5, 0, 1;
   2, 2, 4, 3, 0,  1;
   6, 0, 0, 7, 0,  0, 1;
   4, 2, 0, 5, 4,  0, 0, 1;
   6, 0, 2, 0, 9,  0, 0, 0, 1;
   4, 4, 0, 0, 6,  5, 0, 0, 0, 1;
  10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1;
   4, 2, 2, 2, 0,  7, 6, 0, 0, 0, 0, 1;
   ...

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Theorem 8.4.12 at pp. 246-247.
Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, 1973, p. 37.

Stefano Spezia, First 150 rows of the triangle, flattened
Eric Weisstein's World of Mathematics, Cycle Index.

Cf. A000010, A054523, A130534.

d[n_,k_]:=If[Divisible[n,k],EulerPhi[n/k],0]+If[OddQ[n]&&k==(n+1)/2,n,If[EvenQ[n]&&(k==n/2||k==(n+2)/2),n/2,0]]; Table[d[n,k],{n,3,12},{k,n}]//Flatten (* Stefano Spezia, Jun 26 2023 *)

d(n,k) = A054523(n,k) + d'(n,k), where: If n is odd, then d'(n,k)= n when k=(n+1)/2 and d'(n,k)=0 otherwise. If n is even, then d'(n,k)=n/2 when k=n/2, (n+2)/2 and d'(n,k)=0 otherwise.

Terms corrected by Stefano Spezia, Jun 30 2023

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

Mats Granvik, Oct 12 2023

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

			{
{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
}

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

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}]]

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

Mats Granvik, Oct 12 2023

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

			{
{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
}

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

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}]]

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

A127374 Triangle, row sums = A029935.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 0, 1, 0, 5, 1, 1, 1, 0, 6, 1, 0, 0, 1, 0, 7, 1, 1, 1, 1, 1, 0, 8, 0, 2, 0, 1, 0, 1, 0, 9, 3, 0, 1, 1, 0, 1, 1, 0, 10, 1, 2, 1, 0, 0, 1, 0, 1, 0

Offset: 1

Gary W. Adamson, Jan 12 2007

Row sums = A029935: (1, 2, 4, 5, 8, 8, 12, ...). A127373 = A054521 * A054523

			First few rows of the triangle:
  1;
  2, 0;
  3, 1, 0;
  4, 0, 1, 0;
  5, 1, 1, 1, 0;
  6, 1, 0, 0, 1, 0;
  7, 1, 1, 1, 1, 1, 0;
  8, 0, 2, 0, 1, 0, 1, 0;
  ...

Cf. A054521, A054523, A127373, A029935.

A054523 * A054521 as infinite lower triangular matrices.

A127571 Triangle T(n,k) = phi(n/k)*sigma(k) if k divides n, else 0.

Original entry on oeis.org

1, 1, 3, 2, 0, 4, 2, 3, 0, 7, 4, 0, 0, 0, 6, 2, 6, 4, 0, 0, 12, 6, 0, 0, 0, 0, 0, 8, 4, 6, 0, 7, 0, 0, 0, 15, 6, 0, 8, 0, 0, 0, 0, 0, 13, 4, 12, 0, 0, 6, 0, 0, 0, 0, 18

Offset: 1

Gary W. Adamson, Jan 19 2007

			First few rows of the triangle are:
1;
1, 3;
2, 0, 4;
2, 3, 0, 7;
4, 0, 0, 0, 6;
2, 6, 4, 0, 0, 12;
...

Cf. A000203, A000010, A038040 (row sums), A054523.

T(n,k) = sum_{j=k..n} A054523(n,j)* A130208(j,k), product of the two infinite lower triangular matrices.
T(n,1) = A000010(n).
T(n,n) = A000203(n).

A128980 A054525 * A129691(unsigned).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 29 2007

Row sums = A070777: (1, 1, 2, 1, 4, 2, 6, 1, 2, 4, ...). A129691 = the unsigned inverse of A054523.

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 1;
  0, 0, 0, 1;
  3, 0, 0, 0, 1;
  0, 1, 0, 0, 0, 1;
  5, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 1, 0, 0, 0, 0, 0, 1;
  ...

Cf. A054525, A129691, A070777, A054523.

Moebius transform of A129691

cp's OEIS Frontend

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.

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A127373 Triangle, row sums = A023896, left column = A053570.

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A127465 Triangle read by rows: T(n,k) = k*phi(n/k) if k|n, T(n,k)=0 otherwise.

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A130030 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * prime(d).

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A228094 Triangle starting at row 3 read by rows of the number of permutations in the n-th Dihedral group which are the product of k disjoint cycles, d(n,k), n >= 3, 1 <= k <= n.

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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).

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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).

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A127374 Triangle, row sums = A029935.

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