A128407 -id:A128407 - cp's OEIS frontend

A128408 Triangle read by rows: A128407 * A051731 as infinite lower triangular matrices.

1, -1, -1, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Mar 01 2007

Left and right borders = mu(n), A008683. Row sums = A008966: (1, -2, -2, 0, -2, 4, -2, 0, 0, 4, ...). A128408 * [1,2,3,...] = A063441: (1, -3, -4, 0, -6, 12, ...). A054524 = A051731 * A128407.

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

Cf. A008683, A008966, A054524, A063441, A128407.

a(45) = 0 inserted and more terms from Georg Fischer, Jun 05 2023

A143239 Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 01 2008

Row sums = A000010, phi(n): (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4,...); as a consequence of the Dedekind-Liouville rule illustrated in the example and on p. 137 of "Concrete Mathematics".

			First few rows of the triangle are:
   1;
   2, -1;
   3,  0, -1;
   4, -2,  0,  0;
   5,  0,  0,  0, -1;
   6, -3, -2,  0,  0,  1;
   7,  0,  0,  0,  0,  0, -1;
   8, -4,  0,  0,  0,  0,  0,  0;
   9,  0, -3,  0,  0,  0,  0,  0,  0;
  10, -5,  0,  0, -2,  0,  0,  0,  0,  1;
  11,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
  12, -6, -4,  0,  0,  2,  0,  0,  0,  0,  0,  0;
  13,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
  14, -7,  0,  0,  0,  0, -2,  0,  0,  0,  0,  0,  0,  1;
  ...
Row 12 = (12, -6, -4, 0, 0, 2, 0, 0, 0, 0, 0, 0) since (Cf. A126988 - the divisors of 12 are (12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1) and applying mu(k) * (nonzero terms), we get (1*12, (-1)*6, (-1)*4, 1*2) sum = 4 = phi(12).

Ronald L. Graham, Donald E. Knuth & Oren Patashnik, "Concrete Mathematics" 2nd ed.; Addison-Wesley, 1994, p. 137.

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

Cf. A000010 (row sums), A008683, A126988, A128407.

A143239:= func< n,k | (n mod k) eq 0 select MoebiusMu(k)*(n/k) else 0 >;
[A143239(n,k): k in [1..n], n in [1..14]]; // G. C. Greubel, Sep 12 2024

A143239[n_, k_]:= If[Mod[n,k]==0, MoebiusMu[k]*(n/k), 0];
Table[A143239[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Sep 12 2024 *)

def A143239(n,k): return moebius(k)*(n//k) if (n%k)==0 else 0
flatten([[A143239(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Sep 12 2024

Triangle read by rows generated from the Dedekind-Liouville rule: T(n,k) = mu(k)*(n/k) if k divides n, otherwise T(n,k) = 0 if k is not a divisor of n.

Equals A126988 * A128407.

A176918 Triangle read by rows, a signed variant of A077049 * A128407; as infinite lower triangular matrices.

Original entry on oeis.org

1, -1, 0, -1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 0, -1, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 0, -1, 0, 0

Offset: 1

Gary W. Adamson and Mats Granvik, Apr 29 2010

Row sums = mu(n), A008683

			First few rows of triangle A176918 =
1;
-1, 0;
-1, 0, 0;
-1, 1, 0, 0;
-1, 0, 0, 0, 0;
-1, 1, 1, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0;
-1, 1, 0, 0, 0, 0, 0, 0;
-1, 0, 1, 0, 0, 0, 0, 0, 0;
-1, 1, 0, 0, 1, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
-1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...

Cf. A077049, A128407, A008683, A176890 (another version).

Given (-1)*triangle A077049, preface this with a "1" as row 1; = M.
Perform M * A128407 (the diagonalized variant of A008683); = A176918 as an
infinite lower triangular matrix.

A143256 Triangle read by rows, matrix multiplication A051731 * A128407 * A127648, 1<=k<=n.

Original entry on oeis.org

1, 1, -2, 1, 0, -3, 1, -2, 0, 0, 1, 0, 0, 0, -5, 1, -2, -3, 0, 0, 6, 1, 0, 0, 0, 0, 0, -7, 1, -2, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, -5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 1, -2, -3, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13, 1, -2, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Aug 02 2008

Right border = n*mu(n) = A055615.

Row sums = A023900: (1, -1, -2, -1, -4, 2, -6,...).

			First few rows of the triangle =
1;
1, -2;
1, 0, -3;
1, -2, 0, 0;
1, 0, 0, 0, -5;
1, -2, -3, 0, 0, 6;
1, 0, 0, 0, 0, 0, -7;
...

Robert Israel, Table of n, a(n) for n = 1..9870

Cf. A008683, A055615, A051731, A128407, A127648, A023900.

seq(seq(`if`(i mod j = 0, j*numtheory:-mobius(j),0), j=1..i),i=1..20); # Robert Israel, Sep 07 2014

Table[If[Divisible[n, k], k MoebiusMu[k], 0], {n, 1, 14}, {k, 1, n}] (* Jean-François Alcover, Jun 19 2019 *)

A143256_row = lambda n: [k*moebius(k) if k.divides(n) else 0 for k in (1..n)]
for n in (1..10): print(A143256_row(n)) # Peter Luschny, Jan 05 2018

Triangle read by rows, A051731 * A128407 * A127648, 1<=k<=n

A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 10 2008

The triangle acts as a transform converting any sequence S(k) into a triangle with row sums = S(k). By way of example, begin with S(k), the primes: (2, 3, 5, 7, 11, ...). Add (0, 1, 2, 3, 4, ...) to the sequence getting (prime(n)+(n-1)) = (2, 4, 7, 10, 15, 18, 23, 36, 31, ...) = sequence Q(k). Then replace column 1 (1, 2, 3, ...) of triangle A143349 with sequence Q(k). This = triangle A143350 with row sums prime(n):
    2;
    4, -1;
    7, -1, -1;
   10, -2, -1,  0;
   ...
The A000012 multiplier takes partial sums of A054524 column terms. A051731 is the inverse Mobius transform and A128407 = an infinite lower triangular matrix with mu(n) in the main diagonal and the rest zeros.

			First few rows of the triangle:
   1;
   2, -1;
   3, -1, -1;
   4, -2, -1,  0;
   5, -2, -1,  0, -1;
   6, -3, -2,  0, -1,  1;
   7, -3, -2,  0, -1,  1, -1;
   8, -4, -2,  0, -1,  1, -1,  0;
   9, -4, -3,  0, -1,  1, -1,  0,  0;
  10, -5, -3,  0, -2,  1, -1,  0,  0,  1;
  11, -5, -3,  0, -2,  1, -1,  0,  0,  1, -1;
  12, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0;
  13, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0, -1;
  14, -7, -4,  0, -2,  2, -2,  0,  0,  1, -1,  0, -1,  1;
  ...

Cf. A000012, A051731, A054524, A128407, A143350.

a(39) ff. corrected by Georg Fischer, Jun 05 2023

A143354 Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n.

Original entry on oeis.org

1, 4, -1, 4, 0, -1, 10, -4, 0, 0, 4, 0, 0, 0, -1, 16, -4, -4, 0, 0, 1, 4, 0, 0, 0, 0, 0, -1, 20, -10, 0, 0, 0, 0, 0, 0, 10, 0, -4, 0, 0, 0, 0, 0, 0, 16, -4, 0, 0, -4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 40, -16, -10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 10 2008

Left border = A007426.
Right border = mu(n), A008683.
Row sums = A007425: (1, 3, 3, 6, 3, 9,...).

			First few rows of the triangle =
1;
4, -1;
4, 0, -1;
10, -4, 0, 0;
4, 0, 0, 0, 0, -1;
16, -4, -4, 0, 0, 1;
4, 0, 0, 0, 0, 0, -1;
10, -10, 0, 0, 0, 0, 0, 0;
...

Cf. A051731, A128407, A007425, A007426, A008683.

Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n

A143467 Triangle read by rows, A143315 * A128407, 1<=k<=n.

Original entry on oeis.org

1, 3, -1, 5, 0, -1, 7, -3, 0, 0, 9, 0, 0, 0, -1, 11, -5, -3, 0, 0, 1, 13, 0, 0, 0, 0, 0, -1, 15, -7, 0, 0, 0, 0, 0, 0, 17, 0, -5, 0, 0, 0, 0, 0, 0, 19, -9, 0, 0, -3, 0, 0, 0, 0, 1, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 23, -11, -7, 0, 0, 3, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 17 2008

Row sums = A140434: (1, 2, 4, 4, 8, 4, 12, 8, 12,...).

Right border = mu(n), A008683: (1, -1, -1, 0, -1, 1,...).

			First few rows of the triangle =
1;
3, -1;
5, 0, -1;
7, -3, 0, 0;
9, 0, 0, 0, -1;
11, -5, -3, 0, 0, 1;
13, 0, 0, 0, 0, 0, -1;
...

Cf. A008683, A128407, A143315, A143467.

a(66) corrected by Georg Fischer, Aug 14 2023

A143255 Triangle read by rows, A128407 * A126988; 1<=k<=n.

Original entry on oeis.org

1, -2, -1, -3, 0, -1, 0, 0, 0, 0, -5, 0, 0, 0, -1, 6, 3, 2, 0, 0, 1, -7, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 5, 0, 0, 2, 0, 0, 0, 0, 1, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 14, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 02 2008

Left border = A055615: (1, -2, -3, 0, -5, 6, -7,...).
Right border = A008683, mu(n).
Row sums = A063441: (1, -3, -4, 0, -6, 12, 8,...).

			Triangle begins:
  1;
  -2, -1;
  -3, 0, -1;
  0, 0, 0, 0;
  -5, 0, 0, 0, -1;
  6, 3, 2, 0, 0, 1;
  -7, 0, 0, 0, 0, 0, -1;
  0, 0, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 0, 0;
  10, 5, 0, 0, 2, 0, 0, 0, 0, 1;
  ...

Cf. A008683, A063441, A055615, A128407, A126988.

Cf. A128988.

t[n_, m_] = MoebiusMu[n]*If[m == 1, n, If[Mod[n, m] == 0, n/m, 0]]; Table[t[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Roger L. Bagula, Sep 06 2008 *)

Triangle read by rows, A128407 * A126988; 1<=k<=n.

t(n,m) = MoebiusMu(n) * A126988(n,m); t(n,m) = MoebiusMu(n) * if(m == 1, n, if(n mod m == 0, n/m, 0)). - Roger L. Bagula, Sep 06 2008

A143353 Triangle read by rows, A127172 * A128407 = (A051731)^3 * A128407, 1<=k<=n.

Original entry on oeis.org

1, 3, -1, 3, 0, -1, 6, -3, 0, 0, 3, 0, 0, 0, -1, 9, -3, -3, 0, 0, 1, 3, 0, 0, 0, 0, 0, -1, 10, -6, 0, 0, 0, 0, 0, 0, 6, 0, -3, 0, 0, 0, 0, 0, 0, 9, -3, 0, 0, -3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 18, -9, -6, 0, 0, 3, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 9, -3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 10 2008

Left border = A007425.
Right border = mu(n), A008683.
Row sums = d(n), A000010: (1, 2, 2, 3, 2, 4, 2,...).

			First few rows of the triangle =
1;
3, -1;
3, 0, -1;
6, -3, 0, 0;
3, 0, 0, 0, -1;
9, -3, -3, 0, 0, 1;
3, 0, 0, 0, 0, 0, -1;
10, -6, 0, 0, 0, 0, 0, 0;
6, 0, -3, 0, 0, 0, 0, 0, 0;
9, -3, 0, 0, -3, 0, 0, 0, 0, 1;
...

Cf. A007425, A127172, A128407, A051731, A000010, A008683.

Triangle read by rows, A127172 * A128407 = (A051731)^3 * A128407, 1<=k<=n

A143442 Triangle read by rows, A127648 * A000012 * A128407, 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 15 2008

Row sums = A143443, n * (A002321, the Mertens function) = (1, 0, -3, -4, -10, -6, ...).

Right border = n*mu(n).

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

Cf. A002321, A008683, A127648, A128407, A143443.

Triangle read by rows, A127648 * A000012 * A128407, 1 <= k <= n; where A127648 = an infinite lower triangular matrix with (1, 2, 3, ...) in the main diagonal, the rest zeros. A128407 = diagonalized mu(n) matrix.

cp's OEIS Frontend

A128408 Triangle read by rows: A128407 * A051731 as infinite lower triangular matrices.

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A143239 Triangle read by rows, A126988 * A128407 as infinite lower triangular matrices.

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A176918 Triangle read by rows, a signed variant of A077049 * A128407; as infinite lower triangular matrices.

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A143256 Triangle read by rows, matrix multiplication A051731 * A128407 * A127648, 1<=k<=n.

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A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

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A143354 Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n.

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A143467 Triangle read by rows, A143315 * A128407, 1<=k<=n.

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A143255 Triangle read by rows, A128407 * A126988; 1<=k<=n.

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A143353 Triangle read by rows, A127172 * A128407 = (A051731)^3 * A128407, 1<=k<=n.

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