A054523 -id:A054523 - cp's OEIS frontend

A129691 Inverse of A054523.

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

Offset: 1

Gary W. Adamson, May 04 2007

Row sums = A130054, (1, -1, -1, -3, 0, -5, -2, -3, 0, ...). A129691 * A126988 = A051731. Left column = A023900: (1, -1, -2, -1, -4, 2, -6, ...).

			First few rows of the triangle:
   1;
  -1,  1;
  -2,  0,  1;
  -1, -1,  0,  1;
  -4,  0,  0,  0,  1;
   2, -2, -1,  0,  0,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A054523, A130055, A023900, A126988, A051731.

T(n,k)={if(n%k, 0, sumdivmult(n/k, d, d*moebius(d)))} \\ Andrew Howroyd, Aug 03 2018

A054523^(-1), as an infinite lower triangular matrix.

T(n,k) = A023900(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

Terms a(56) and beyond from Andrew Howroyd, Aug 03 2018

A127649 A127648 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 6, 0, 3, 8, 4, 0, 4, 20, 0, 0, 0, 5, 12, 12, 6, 0, 0, 6, 42, 0, 0, 0, 0, 0, 7, 32, 16, 0, 8, 0, 0, 0, 8, 54, 0, 18, 0, 0, 0, 0, 0, 9, 40, 40, 0, 0, 10, 0, 0, 0, 0, 10, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 48, 24, 24, 24, 0, 12, 0, 0, 0, 0, 0, 12, 156, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 84

Offset: 1

Gary W. Adamson, Jan 22 2007

Natural number transform of A054523.

Row sums = n^2, left column = A002618

			First few rows of the triangle are:
1;
2, 2;
6, 0, 3;
8, 4, 0, 4;
20, 0, 0, 0, 5;
12, 12, 6, 0, 0, 6;
42, 0, 0, 0, 0, 0, 7;
...

Cf. A127648, A002618, A054523.

A054523 := proc(n,k) if n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi ; end: A127649 := proc(n,k) A054523(n,k)*n ; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127649(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007

T(n,k)=n*A054523(n,k). - R. J. Mathar, Nov 01 2007

T(n,k) = Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y), n) = k], where f(x,y) = x - y. - Mats Granvik, Oct 08 2023

More terms from R. J. Mathar, Nov 01 2007

A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

Original entry on oeis.org

1, 2, 1, 3, 0, 2, 4, 2, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 4, 0, 4, 0, 0, 0, 4, 9, 0, 6, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 6, 8, 6, 0, 4, 0, 0, 0, 0, 0, 4, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 14, 7, 0, 0, 0, 0, 12, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054523 and A054522 are commuted, the matrix product becomes A127477.

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

Cf. A054522, A054523, A018804, A000010.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127478 := proc(n,k) add( A054523(n,j)*A054522(j,k), j=k..n) ; end: seq(seq( A127478(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054523(n,j) * A054522(j,k).
T(n,n) = A000010(n) (diagonal).
sum_{k=1..n} T(n,k) = A018804(n) (row sums).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

A129559 A054523 * A115361.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 4, 2, 0, 1, 4, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 8, 4, 0, 0, 2, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 4, 4, 2, 0, 2, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 20 2007

row sums = A129527: (1, 3, 3, 7, 5, 9, 7, 15, ...). Left column = phi(2*n), A062570: (1, 2, 2, 4, 4, 4, 6, 8, ...).

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

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A062570.
Row sums are A129527 (inverse Moebius transform of A062570).
Cf. A054523, A115361.

T(n, k)=if(n%k, 0, eulerphi(2*n/k)) \\ Andrew Howroyd, Aug 07 2018

Equals A054523 * A115361 as infinite lower triangular matrices.

T(n,k) = phi(2*n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 07 2018

Terms a(56) and beyond from Andrew Howroyd, Aug 07 2018

A127192 Triangle read by rows: square of A054523.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 5, 2, 0, 1, 8, 0, 0, 0, 1, 8, 4, 2, 0, 0, 1, 12, 0, 0, 0, 0, 0, 1, 12, 5, 0, 2, 0, 0, 0, 1, 16, 0, 4, 0, 0, 0, 0, 0, 1, 16, 8, 0, 0, 2, 0, 0, 0, 0, 1, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 8, 5, 4, 0, 2, 0, 0, 0, 0, 0, 1, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 07 2007

Row sums = A018804: (1, 3, 5, 8, 9, 15, ...), sum of gcd(k,n) for 1<= k <= n. Left column = A029935: (1, 2, 4, 5, 8, 8, 12, 12, ...). A127192 * d(n) = d(n) * n, or A127192 * A000005 = A038040 = (1, 4, 6, 12, 10, 24, 14, ...).

Column k contains terms of A029935 interspersed with (k-1) zeros.

			First few rows of the triangle:
   1;
   2, 1;
   4, 0, 1;
   5, 2, 0, 1;
   8, 0, 0, 0, 1;
   8, 4, 2, 0, 0, 1;
  12, 0, 0, 0, 0, 0, 1;
  12, 5, 0, 2, 0, 0, 0, 1;
  16, 0, 4, 0, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A018804, A029935, A038040, A054523.

a(20) ff. corrected and more terms from Georg Fischer, May 31 2023

A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

Original entry on oeis.org

1, 2, 1, 5, 0, 2, 6, 3, 0, 2, 17, 0, 0, 0, 4, 10, 5, 4, 0, 0, 2, 37, 0, 0, 0, 0, 0, 6, 22, 11, 0, 6, 0, 0, 0, 4, 41, 0, 14, 0, 0, 0, 0, 0, 6, 34, 17, 0, 0, 8, 0, 0, 0, 0, 4, 101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 30, 15, 12, 10, 0, 6, 0, 0, 0, 0, 0, 4, 145, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 74, 37, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

If the two matrices A054522 and A054523 are commuted, the matrix product becomes A127478.

			First few rows of the triangle are:
1;
2, 1;
5, 0, 2;
6, 3, 0, 2;
17, 0, 0, 0, 4;
10, 5, 4, 0, 0, 2;
37, 0, 0, 0, 0, 0, 6;
22, 11, 0, 6, 0, 0, 0, 4;

Cf. A054522, A054523, A057660, A000010, A029939.

A054522 := proc(n,k) if k = 1 then 1; elif n mod k = 0 then numtheory[phi](k) ; else 0 ; fi; end:
A054523 := proc(n,k) if k = n then 1; elif n mod k = 0 then numtheory[phi](n/k) ; else 0 ; fi; end:
A127477 := proc(n,k) add( A054522(n,j)*A054523(j,k), j=k..n) ; end: seq(seq( A127477(n,k),k=1..n),n=1..15) ;

T(n,k) = sum_{j=k..n} A054522(n,j) * A054523(j,k).
sum_{k=1..n} T(n,k) = A057660(n) (row sums).
T(n,n) = A000010(n) (diagonal).
T(n,1) = A029939(n).

Converted comments to formulas, extended - R. J. Mathar, Sep 11 2009

A128391 A054523^24 * A000594.

Original entry on oeis.org

1, 0, 300, -1724, 4926, 0, -16600, 44600, -100299, 0, 534852, -517200, -577450, 0, 1477800, 2486626, -6905550, 0, 10661852, -8492424, -4980000, 0, 18643800, 13380000, -25030649, 0, -78396200, 28618400, 128407302, 0, -52842448, -150834520, 160455600, 0, -81771600

Offset: 1

Gary W. Adamson, Feb 28 2007

Conjecture: Given A054523^k, k = any positive integer, "zero" appears only in the sequence A018391 (k=24).

Each application of A054523 corresponds to the Dirichlet convolution of A000010 with the sequence on the right. Since both A000594 and A000010 are multiplicative, the resulting sequence will also be multiplicative. - Andrew Howroyd, Aug 03 2018

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000010, A000594, A054523, A128378, A128379, A128380, A128381, A128392.

nmax = 40;
T[n_, k_] := If[Divisible[n, k], EulerPhi[n/k], 0]; T[1, 1] = 1;
M = Table[T[n, k], {n, 1, nmax}, {k, 1, nmax}];
MatrixPower[M, 24].RamanujanTau[Range[nmax]] (* Jean-François Alcover, Sep 20 2019 *)

seq(n, k=24)={my(u=vector(n,n,eulerphi(n)), v=vector(n,n,ramanujantau(n))); for(i=1, k, v=dirmul(u,v)); v} \\ Andrew Howroyd, Aug 03 2018

A054523^24 as an infinite lower triangular matrix * A000594.

a(7) corrected and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 4, 0, 0, 1, 1, 4, 3, 1, 0, 1, 1, 6, 0, 0, 0, 0, 1, 1, 6, 2, 1, 1, 0, 0, 1, 1, 6, 2, 2, 0, 0, 0, 0, 1, 1, 8, 4, 0, 1, 1, 0, 0, 0, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

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

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000010 (alternating row sums), A053158 (row sums).
Cf. A000038, A019590, A054523, A054977.
Cf. A063524, A097806, A126246, A183918.

A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
[A129479(n,k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024

A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[kA054523[n, j+k], {j,0,1}], 1];
Table[T[n,k],{n,16},{k,n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)

def A054523(n,k):
    if (k==n): return 1
    elif (n%k): return 0
    else: return euler_phi(n//k)
def A129479(n, k):
    if k<0 or k>n: return 0
    elif k==n: return 1
    else: return A054523(n,k) + A054523(n,k+1)
flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, Feb 11 2024

Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
T(n, 1) = A126246(n).
From G. C. Greubel, Feb 11 2024: (Start)
T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
T(2*n-1, n) = A019590(n).
T(2*n, n) = A054977(n).
T(2*n+1, n) = A000038(n).
T(3*n, n) = A063524(n-1).
T(3*n-2, n) = A183918(n+2).
Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)

A126705 A097806 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 1, 1, 6, 1, 0, 1, 1, 6, 2, 1, 0, 1, 1, 8, 2, 1, 0, 0, 1, 1, 10, 2, 0, 1, 0, 0, 1, 1, 10, 2, 2, 1, 0, 0, 0, 1, 1, 10, 4, 2, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

Row sums = (1, 3, 5, 7, 9, ...). A129479 = A054523 * A097806. A097806 = the pairwise operator.

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

Cf. A097806 * A054523, A129479.

A128317 Triangle read by rows: T = A054523 * A130595, as a lower triangular matrix.

Original entry on oeis.org

1, 0, 1, 3, -2, 1, 0, 4, -3, 1, 5, -4, 6, -4, 1, 0, 5, -9, 10, -5, 1, 7, -6, 15, -20, 15, -6, 1, 0, 12, -24, 36, -35, 21, -7, 1, 9, -12, 30, -56, 70, -56, 28, -8, 1, 0, 9, -30, 80, -125, 126, -84, 36, -9, 1, 11, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

			First few rows of the triangle:
  1;
  0,  1;
  3, -2,  1;
  0,  4, -3,   1;
  5, -4,  6,  -4,  1;
  0,  5, -9,  10, -5,  1;
  7, -6, 15, -20, 15, -6, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000984, A001700, A007318, A054523, A130595.

Sums include: A000010 (row sums), A126246.

A128317:= func< n,k | (&+[(-1)^(d+k)*EulerPhi(Floor(n/d))*Binomial(d-1, k-1) : d in Divisors(n)]) >;
[A128317(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

A128317[n_, k_]:= DivisorSum[n, (-1)^(#+k)*EulerPhi[n/#]*Binomial[#-1, k-1]  &];
Table[A128317[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 24 2024 *)

def A128317(n,k): return sum((-1)^(k+j)*euler_phi(n/j)*binomial(j-1, k-1) for j in (1..n) if (j).divides(n))
flatten([[A128317(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 24 2024

Equals A054523 * signed A007318 as infinite lower triangular matrices. A007318 is signed by columns: (+, -, +, ...).
Sum_{k=1..n} T(n, k) = A000010(n) (row sums).
From G. C. Greubel, Jun 24 2024: (Start)
T(n, k) = A054523 * A130595, as a lower triangular matrix.
T(n, k) = Sum_{j=k..n} (-1)^(k+j)*A054523(n,j)*binomial(j-1, k-1).
T(n, k) = Sum_{d|n} (-1)^(d+k)*EulerPhi(n/d)*binomial(d-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
T(2*n-2, n-1) = (-1)^n*A001700(n-2), n >= 2.
Sum_{k=1..n} k*T(n, k) = A126246(n). (End)

cp's OEIS Frontend

A129691 Inverse of A054523.

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A127649 A127648 * A054523 as infinite lower triangular matrices.

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A127478 Triangle T(n,k) read by rows: matrix product A054523 * A054522.

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A129559 A054523 * A115361.

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A127192 Triangle read by rows: square of A054523.

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A127477 Triangle T(n,k) read by rows: matrix product A054522 * A054523.

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A128391 A054523^24 * A000594.

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A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

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