A126988 -id:A126988 - cp's OEIS frontend

A127013 Triangle read by rows: row reversal of A126988.

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

Offset: 1

Gary W. Adamson, Jan 02 2007

Let j = reversed indices of row terms. Then for any row, j*T(n,k) = n, for nonzero T(n,k). For example, in row 10, we match the terms with their j indices: (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), (dot product) (10, 9, 8, 7, 6, 5, 4, 3, 2, 1); getting (10, 0, 0, 0, 0, 10, 0, 0, 10, 10).
The factors of n are found in each row in order, as nonzero terms; e.g., 10 has the factors 1, 2, 5, 10, sum 18.
Row sums = sigma(n), A000203.

			First few rows of the triangle are:
   1;
   1, 2;
   1, 0, 3;
   1, 0, 2, 4;
   1, 0, 0, 0, 5;
   1, 0, 0, 2, 3, 6;
   1, 0, 0, 0, 0, 0, 7;
   1, 0, 0, 0, 2, 0, 4, 8;
   1, 0, 0, 0, 0, 0, 3, 0, 9;
   1, 0, 0, 0, 0, 2, 0, 0, 5, 10;
Row 10 = (1, 0, 0, 0, 0, 2, 0, 0, 5, 10), reversal of 10th row of A126988.

David Wells, "Prime Numbers, The Most Mysterious Figures in Math", John Wiley & Sons, 2005, Appendix.

Reinhard Zumkeller, Table of n, a(n) for n = 1..7875

Cf. A126988, A000203.

a127013 n k = a127013_tabl !! (n-1) !! (k-1)
a127013_row n = a127013_tabl !! (n-1)
a127013_tabl = map reverse a126988_tabl
-- Reinhard Zumkeller, Jan 20 2014

[[(n mod (n-k+1)) eq 0 select n/(n-k+1) else 0: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 03 2019

T[n_,m_]:= If[Mod[n, m]==0, n/m, 0]; Table[T[n,n-m+1], {n, 1, 12}, {m, 1, n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

{T(n, k) = if(n%k==0, n/k, 0)};
for(n=1,12, for(k=1,n, print1(T(n,n-k+1), ", "))) \\ G. C. Greubel, Jun 03 2019

def T(n, k):
    if (n%k==0): return n/k
    else: return 0
[[T(n, n-k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 03 2019

T(10,10) fixed by Reinhard Zumkeller, Jan 20 2014

More terms added by G. C. Greubel, Jun 03 2019

A127097 Triangle T(n,m) = A127093 * A126988 read by rows.

Original entry on oeis.org

1, 5, 2, 10, 0, 3, 21, 10, 0, 4, 26, 0, 0, 0, 5, 50, 20, 15, 0, 0, 6, 50, 0, 0, 0, 0, 0, 7, 85, 42, 0, 20, 0, 0, 0, 8, 91, 0, 30, 0, 0, 0, 0, 0, 9, 130, 52, 0, 0, 25, 0, 0, 0, 0, 10, 122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 210, 100, 63, 40, 0, 30, 0, 0, 0, 0, 0, 12, 170, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A127093 and A126988.

			First few rows of the triangle are:
1;
5, 2;
10, 0, 3;
21, 10, 0, 4;
26, 0, 0, 0, 5;
50, 20, 15, 0, 0, 6;
50, 0, 0, 0, 0, 0, 7;
...

Cf. A126988, A000203, A127098, A001187, A001001 (row sums), A127093.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A126988 := proc(n,k) if n mod k = 0 then n/k; else 0; fi; end:
A127097 := proc(n,m) add( A127093(n,j)*A126988(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127097(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T(n,m) = sum_{j=m..n} A127093(n,j)*A126988(j,m).

T(n,1) = A001157(n).

A-numbers corrected by R. J. Mathar, Aug 18 2009

A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

Original entry on oeis.org

1, 3, 2, 4, 0, 3, 7, 6, 0, 4, 6, 0, 0, 0, 5, 12, 8, 9, 0, 0, 6, 8, 0, 0, 0, 0, 0, 7, 15, 14, 0, 12, 0, 0, 0, 8, 13, 0, 12, 0, 0, 0, 0, 0, 9, 18, 12, 0, 0, 15, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 28, 24, 21, 16, 0, 18, 0, 0, 0, 0, 0, 12, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 24, 16, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A126988 and A127093.

			First few rows of the triangle are:
1;
3, 2;
4, 0, 3;
7, 6, 0, 4;
6, 0, 0, 0, 5;
12, 8, 9, 0, 0, 6;
8, 0, 0, 0, 0, 0, 7;
15, 14, 0, 12, 0, 0, 0, 8;
13, 0, 12, 0, 0, 0, 0, 0, 9;
18, 12, 0, 0, 15, 0, 0, 0, 0, 10;
...

Cf. A060640 (row sums), A000203, A127093, A127094, A123229, A127096, A127097, A127098, A127013, A127057.

T(n,m) = sum_{j=m..n} A126988(n,j)*A127093(j,m).

T(n,1) = A000203(n).

Extended by R. J. Mathar, Aug 18 2009

A143315 Triangle read by rows: T(n, k) = 2*A126988(n, k) - signum(A126988(n, k)).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A129235: (1, 4, 6, 11, 10, 20, 14,...).

			First few rows of the triangle:
   1;
   3, 1;
   5, 0, 1;
   7, 3, 0, 1;
   9, 0, 0, 0, 1;
  11, 5, 3, 0, 0, 1;
  13, 0, 0, 0, 0, 0, 1;
  15, 7, 0, 3, 0, 0, 0, 1;
  ...

Cf. A051731, A126988, A129235.

By columns, replace the 1's in A051731 in succession with (1, 3, 5, 7,...).

Definition corrected and a(50) split by Georg Fischer, Jun 08 2023

A127139 Inverse triangle of A126988.

Original entry on oeis.org

1, -2, 1, -3, 0, 1, 0, -2, 0, 1, -5, 0, 0, 0, 1, 6, -3, -2, 0, 0, 1, -7, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 0, 1, 10, -5, 0, 0, -2, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Row sums give A023900.
Left column is A055615.
A127139 * [1, 2, 3, ...] = [1, 0, 0, 0, ...].
A127139 * [1, 0, 0, 0, ...] = A055615.
A127140 is the square of A127139.

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

Cf. A126988, A055615, A023900, A127140.

nn = 10; s = 0; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, -Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Mar 12 2016 *)

Inverse triangle of A126988.

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.

A128392 A126988^12 * A000594.

Original entry on oeis.org

1, 0, 288, -1736, 4890, 0, -16660, 44576, -103869, 0, 534744, -499968, -577582, 0, 1408320, 2507344, -6905730, 0, 10661648, -8489040, -4798080, 0, 18643548, 12837888, -25207475, 0, -77183496, 28921760, 128406978, 0, -52842796, -151328128, 154006272, 0, -81467400

Offset: 1

Gary W. Adamson, Feb 28 2007

Conjecture: Given A126988^k, k any positive integer, A128392 is the only sequence in the infinite set with zeros.

Each application of A126988 corresponds to the Dirichlet convolution of the natural numbers with the sequence on the right. Since both Ramanujan's tau function A000594 and the natural numbers 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. A126988, A000594, A128378, A128379, A128380, A128381, A128391.

nmax = 40;
M = Table[If[Mod[n, m] == 0, n/m, 0], {n, 1, nmax}, {m, 1, nmax}];
MatrixPower[M, 12].RamanujanTau[Range[nmax]] (* Jean-François Alcover, Sep 20 2019 *)

seq(n, k=12)={my(u=vector(n,n,n), v=vector(n,n,ramanujantau(n))); for(i=1, k, v=dirmul(u,v)); v} \\ Andrew Howroyd, Aug 03 2018

A126988 as an infinite lower triangular matrix, * A000594.

Terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

A128489 Triangle read by rows: A000012 * A126988 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 6, 1, 1, 10, 3, 1, 1, 15, 3, 1, 1, 1, 21, 6, 3, 1, 1, 1, 28, 6, 3, 1, 1, 1, 1, 36, 10, 3, 3, 1, 1, 1, 1, 45, 10, 6, 3, 1, 1, 1, 1, 1, 55, 15, 6, 3, 3, 1, 1, 1, 1, 1, 66, 15, 6, 3, 3, 1, 1, 1, 1, 1, 1, 78, 21, 10, 6, 3, 3, 1, 1, 1, 1, 1, 1, 91, 21, 10, 6, 3, 3, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Mar 04 2007

Row sums = A024916: (1, 4, 8, 15, 21, 33, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  1, 1;
  10,  3, 1, 1;
  15,  3, 1, 1, 1;
  21,  6, 3, 1, 1, 1;
  28,  6, 3, 1, 1, 1, 1;
  36, 10, 3, 3, 1, 1, 1, 1;
  45, 10, 6, 3, 1, 1, 1, 1, 1;
  ...

Cf. A000012, A000217, A024916, A126988.

By columns, k=1,2,3,...; k repeated terms of the triangular series, (1, 3, 6, 10, ...) in the k-th column.

a(11) = 1 inserted and more terms from Georg Fischer, May 29 2023

A143311 Triangle read by rows, A127648 * A126988; 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 9, 0, 3, 16, 8, 0, 4, 25, 0, 0, 0, 5, 36, 18, 12, 0, 0, 6, 49, 0, 0, 0, 0, 0, 7, 64, 32, 0, 16, 0, 0, 0, 8, 81, 0, 27, 0, 0, 0, 0, 0, 9, 100, 50, 0, 0, 20, 0, 0, 0, 0, 10, 121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 144, 72, 48, 36, 0, 24, 0, 0, 0, 0, 0, 12, 169, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A064987, n*sigma(n): (1, 6, 12, 28, 30, 72, 56,...).

			First few rows of the triangle =
1;
4, 2;
9, 0, 3;
16, 8, 0, 4;
25, 0, 0, 0, 5;
36, 18, 12, 0, 0, 6;
49, 0, 0, 0, 0, 0, 7;
64, 32, 0, 16, 0, 0, 0, 8;
...

Cf. A127648, A126988, A064987.

Triangle read by rows, A127648 * A126988; 1<=k<=n

A167990 Elements in A126988 (by row) that are not 1.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Nov 16 2009

GCD of rows is A014963.

			Table begins:
   0;
   2, 0;
   3, 0, 0;
   4, 2, 0, 0;
   5, 0, 0, 0, 0;
   6, 3, 2, 0, 0, 0;
   7, 0, 0, 0, 0, 0, 0;
   8, 4, 0, 2, 0, 0, 0, 0;
   9, 0, 3, 0, 0, 0, 0, 0, 0;
  10, 5, 0, 0, 2, 0, 0, 0, 0, 0;
  11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 0;

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

Cf. A014963, A039653, A126988.

[k eq n or (n mod k) ne 0 select 0 else n/k: k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 13 2023

Table[If[k==n || Mod[n, k]!=0, 0, n/k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 13 2023 *)

def A167990(n, k):
    if (k==n or n%k!=0): return 0
    else: return n/k
flatten([[A167990(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jan 13 2023

From G. C. Greubel, Jan 13 2023: (Start)
T(n, k) = 0 if k = n or (n mod k) != 0, otherwise T(n, k) = n/k.
T(n, 1) = n - [n=1].
T(m*n, n) = m, m >= 2.
Sum_{k=1..n} T(n, k) = A039653(n). (End)

cp's OEIS Frontend

A127013 Triangle read by rows: row reversal of A126988.

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A127097 Triangle T(n,m) = A127093 * A126988 read by rows.

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A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

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A143315 Triangle read by rows: T(n, k) = 2*A126988(n, k) - signum(A126988(n, k)).

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A127139 Inverse triangle of A126988.

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

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A128392 A126988^12 * A000594.

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