A127093 -id:A127093 - cp's OEIS frontend

A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

1, 1, 1, 2, 2, 1, 2, 1, 3, 3, 2, 4, 1, 3, 1, 2, 4, 5, 3, 6, 2, 6, 1, 2, 4, 1, 5, 7, 5, 10, 3, 9, 2, 4, 8, 1, 5, 1, 2, 3, 6, 11, 7, 14, 5, 15, 3, 6, 12, 2, 10, 1, 2, 3, 6, 1, 7, 15, 11, 22, 7, 21, 5, 10, 20, 3, 15, 2, 4, 6, 12, 1, 7, 1, 2, 4, 8, 22, 15, 30, 11, 33, 7, 14, 28, 5, 25

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A340035.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  [1];
  [1],  [1, 2];
  [2],  [1, 2],  [1, 3];
  [3],  [2, 4],  [1, 3],  [1, 2, 4];
  [5],  [3, 6],  [2, 6],  [1, 2, 4],  [1, 5];
  [7],  [5, 10], [3, 9],  [2, 4, 8],  [1, 5],  [1, 2, 3, 6];
  [11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
  ...
Row sums gives A066186.
Written as a tetrahedrons the first five slices are:
  --
  1;
  --
  1,
  1, 2;
  -----
  2,
  1, 2,
  1, 3;
  -----
  3,
  2, 4,
  1, 3,
  1, 2, 4;
  --------
  5,
  3, 6,
  2, 6,
  1, 2, 4,
  1, 5;
  --------
Row sums give A221529.
The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   |    -    |     |       |         |           |  5          |
| C |    -    |     |       |         |  3        |  3 6        |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| D | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
| D | A027750 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A027750 |     |       |         |  1        |  1 2        |
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A027750 |     |       |  1      |  1 2      |  1   3      |
| S | A027750 |     |       |  1      |  1 2      |  1   3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340056 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11528 (rows 1..75 of the triangle, flattened)

Nonzero terms of A221649.

Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m],{m,n}]];Array[A340057row,10] (* Paolo Xausa, Sep 02 2023 *)

A132442 Triangle whose n-th row consists of the first n terms of the n-th row of A134866.

Original entry on oeis.org

1, 1, 3, 1, 1, 4, 1, 3, 1, 7, 1, 1, 1, 1, 6, 1, 3, 4, 3, 1, 12, 1, 1, 1, 1, 1, 1, 8, 1, 3, 1, 7, 1, 3, 1, 15, 1, 1, 4, 1, 1, 4, 1, 1, 13, 1, 3, 1, 3, 6, 3, 1, 3, 1, 18, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 3, 4, 7, 1, 12, 1, 7, 4, 3, 1, 28, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 3, 1, 3, 1, 3, 8, 3, 1, 3, 1, 3, 1, 24

Offset: 1

Gary W. Adamson, Nov 14 2007

Previous name: Triangle, n-th row = first n terms of n-th row of an array formed by A051731 * A127093 (transform).
Right border = sigma(n), A000203.
Row sums = A038040.
The function T(n,k) = T(k,n) is defined for k > n, but only the values of k in 1..n as a triangular array are listed here.

			First few rows of the A134866 array:
  1,  1,  1,  1,  1,  1,  1, ...
  1,  3,  1,  3,  1,  3,  1, ...
  1,  1,  4,  1,  1,  4,  1, ...
  1,  3,  1,  7,  1,  3,  1, ...
  1,  1,  1,  1,  6,  1,  1, ...
  1,  3,  4,  3,  1, 12,  1, ...
  ...
First few rows of the triangle:
  1;
  1,  3;
  1,  1,  4;
  1,  3,  1,  7;
  1,  1,  1,  1,  6;
  1,  3,  4,  3,  1, 12;
  1,  1,  1,  1,  1,  1,  8;
  1,  3,  1,  7,  1,  3,  1, 15;
  ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A051731, A127093.
Cf. A038040 (row sums), A000203 (right border), A050873 (gcd(n,k)).
Cf. A000142 (determinant).
Cf. A134866.

a132442 n k = a132442_tabl !! (n-1) !! (k-1)
a132442_row n = a132442_tabl !! (n-1)
a132442_tabl = map (map a000203) a050873_tabl
-- Reinhard Zumkeller, Dec 12 2015

T[ n_, k_] := If[ n < 1 || k < 1, 0, If[ k > n, T[ k, n], If[ k == 1, 1, If[ n > k, T[ k, Mod[ n, k, 1]],  DivisorSigma [1, n]]]]] (* Michael Somos, Jul 18 2011 *)

{T(n, k) = if( n<1 || k<1, 0, if( k>n, T(k, n), if( k==1, 1, if( n>k, T(k, (n-1)%k+1), sigma( n)))))} /* Michael Somos, Jul 18 2011 */

T(n,k) = A000203(gcd(n,k)). - Reinhard Zumkeller, Dec 12 2015

Missing T(10,9) = 1 inserted by Reinhard Zumkeller, Dec 12 2015

Name edited by Michel Marcus, Dec 21 2022

A134866 Table read by antidiagonals: T(n,k) = sigma(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 3, 1, 3, 6, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 7, 1, 12, 1, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 8, 3, 1, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

Previous name was: Triangle, antidiagonals of an array formed by A051731 * A127093 (transform).

Row sums give A094471.

			First few rows of the array:
  1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 3, 1, 3, 1, ...
  1, 1, 4, 1, 1, 4, 1, ...
  1, 3, 1, 7, 1, 3, 1, ...
  1, 1, 1, 1, 6, 1, 1, ...
  ...
First antidiagonals:
  1;
  1, 1;
  1, 3, 1;
  1, 1, 1, 1;
  1, 3, 4, 3, 1;
  1, 1, 1, 1, 1, 1;
  1, 3, 1, 7, 1, 3, 1;
  1, 1, 4, 1, 1, 4, 1, 1;
  1, 3, 1, 3, 6, 3, 1, 3, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Cf. A051731, A127093, A094471, A132442.

Table[DivisorSigma[1, GCD[#, k]] &[n - k + 1], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 19 2022 *)

T(n, k) = sigma(gcd(n, k)); \\ Michel Marcus, Dec 19 2022

T(n,k) = A000203(A050873(n,k)). - Michel Marcus, Dec 19 2022

New name and data corrected by Michel Marcus, Dec 19 2022

A173541 Triangle read by rows: T(n,k)=k if k is a proper non-divisor of n, otherwise T(n,k)=0 (1<=k<=n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 25 2010

Observation: Note that the k-th column is a sequence where the periodic subset is formed by zero together with k-1 numbers k. For example, the 5th column can be defined as "Period 5: repeat 0,5,5,5,5".

			Triangle begins:
0;
0,0;
0,2,0;
0,0,3,0;
0,2,3,4,0;
0,0,0,4,5,0;
0,2,3,4,5,6,0;
0,0,3,0,5,6,7,0;
0,2,0,4,5,6,7,8,0;
0,0,3,4,0,6,7,8,9,0;
0,2,3,4,5,6,7,8,9,10,0;
0,0,0,0,5,0,7,8,9,10,11,0;
0,2,3,4,5,6,7,8,9,10,11,12,0;
0,0,3,4,5,6,0,8,9,10,11,12,13,0;
0,2,0,4,0,6,7,8,9,10,11,12,13,14,0;
0,0,3,0,5,6,7,0,9,10,11,12,13,14,15,0;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A049820, A127093, A173540. Row sums give A024816.

Cf. A002260.

a173541 n k = a173541_tabl !! (n-1) !! (n-1)
a173541_row n = a173541_tabl !! (n-1)
a173541_tabl = zipWith (zipWith (*))
                       a002260_tabl $ map (map (1 -)) a051731_tabl
-- Reinhard Zumkeller, Feb 19 2014

T(n,k) = k * A051731(n,k). - Reinhard Zumkeller, Feb 19 2014

A228812 Triangle read by rows: T(n,k), n>=1, k>=1, in which row n lists m terms, where m = A055086(n). If k divides n and k < n^(1/2) then T(n,k) = k and T(n,m-k+1) = n/T(n,k). Also, if k^2 = n then T(n,k) = k. Other terms are zeros.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 0, 5, 1, 2, 3, 6, 1, 0, 0, 7, 1, 2, 4, 8, 1, 0, 3, 0, 9, 1, 2, 0, 5, 10, 1, 0, 0, 0, 11, 1, 2, 3, 4, 6, 12, 1, 0, 0, 0, 0, 13, 1, 2, 0, 0, 7, 14, 1, 0, 3, 5, 0, 15, 1, 2, 0, 4, 0, 8, 16, 1, 0, 0, 0, 0, 0, 17, 1, 2, 3, 0, 6, 9, 18

Offset: 1

Omar E. Pol, Oct 03 2013

The number of positive terms of row n is A000005(n).
The positive terms of row n are the divisors of n in increasing order.
Row n has length A055086(n).
Column k starts in row A002620(k+1).
The number of zeros in row n equals A078152(n).
The sum of row n is A000203(n).
Positive terms give A027750.
It appears that there are only eight rows that do not contain zeros. The indices of these rows are 1, 2, 3, 4, 6, 8, 12, 24, the divisors of 24, see A018253.
For another version see A228814.

			For n = 60 the 60th row of triangle is [1, 2, 3, 4, 5, 6, 0, 0, 10, 12, 15, 20, 30, 60]. The row length is A055086(60) = 14. The number of zeros is A078152(60) = 2. The number of positive terms is A000005(60) = 12. The positive terms are the divisors of 60. The row sum is A000203(60) = 168.
Triangle begins:
1;
1,  2;
1,  3;
1,  2,  4;
1,  0,  5;
1,  2,  3,  6;
1,  0,  0,  7;
1,  2,  4,  8;
1,  0,  3,  0,  9;
1,  2,  0,  5, 10;
1,  0,  0,  0, 11;
1,  2,  3,  4,  6, 12;
1,  0,  0,  0,  0, 13;
1,  2,  0,  0,  7, 14;
1,  0,  3,  5,  0, 15;
1,  2,  0,  4,  0,  8, 16;
1,  0,  0,  0,  0,  0, 17;
1,  2,  3,  0,  6,  9, 18;
1,  0,  0,  0,  0,  0, 19;
1,  2,  0,  4,  5,  0, 10, 20;
1,  0,  3,  0,  0,  7,  0, 21;
1,  2,  0,  0,  0,  0, 11, 22;
1,  0,  0,  0,  0,  0,  0, 23;
1,  2,  3,  4,  6,  8, 12, 24;
...

Column 1 is A000012.
Right border gives A000027.
Cf. A000005, A000203, A002620, A004526, A018253, A027750, A055086, A078152, A127093, A147861, A161904, A196020, A210959, A212119, A212120, A221645, A228813, A228814.

A334947 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 6, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th octagonal number (A000567).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 0

Offset: 1

Omar E. Pol, May 27 2020

Since the trivial partition n is counted, so T(n,1) = 1.
This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th octagonal number.
This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.
For a general theorem about the triangles of this family see A285914.

			Triangle begins (rows 1..24).
1;
1;
1;
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
...
For n = 24 there are three partitions of 24 into consecutive parts that differ by 6, including 24 as a valid partition. They are [24], [15, 9] and [14, 8, 2]. There are 1, 2 and 3 parts respectively, so the 24th row of this triangle is [1, 2, 3].

Row sums give A334949.
Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), A334462 (d=4), A334540 (d=5), this sequence (d=6).
Cf. A000567, A334946, A334948, A334953.

T(n,k) = k*A334946(n,k).

A143112 A051731 * A032742 = sum of largest proper divisors of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 5, 8, 2, 14, 2, 10, 8, 16, 2, 18, 2, 20, 10, 14, 2, 30, 7, 16, 14, 26, 2, 32, 2, 32, 14, 20, 10, 44, 2, 22, 16, 44, 2, 42, 2, 38, 26, 26, 2, 62, 9, 38, 20, 44, 2, 54, 14, 58, 22, 32, 2, 80, 2, 34, 34, 64, 16, 62, 2, 56, 26, 58, 2, 96, 2, 40, 38, 62, 14, 72, 2, 92

Offset: 1

Gary W. Adamson and Mats Granvik, Jul 25 2008

nonn

Inverse Möbius transform of A032742. - Antti Karttunen, Sep 25 2018

			a(12) = 14. The divisors of 12 are shown in row 12 of triangle A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12). The largest proper divisors of these terms are (1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6), sum = 14. Or, we can take row of triangle A051731: (1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1) dot (1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6) = (1 + 1 + 1 + 2 + 0 + 3 + 0 + 0 + 0 + 0 + 0 + 6) = 14, where A032742 = (1, 1, 1, 2, 1, 3, 1, 4, 3, 5,...).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A051731, A127093, A032742, A300236, A305807, A305808.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A143112(n) = sumdiv(n,d,A032742(d)); \\ Antti Karttunen, Sep 25 2018

A051731 * A032742, where A051731 = the inverse Mobius transform and A032742 = the largest proper divisors of n: (1, 1, 1, 3, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7,...).

a(n) = Sum_{d|n} A032742(d). - Antti Karttunen, Sep 25 2018

More terms from R. J. Mathar, Jan 19 2009

A143343 Triangle T(n,k) (n>=0, 1<=k<=n+1) read by rows: T(n,1)=1 for n>=0, T(1,2)=2. If n>=3 is odd then T(n,k)=1 for all k. If n>=3 is even then if k is prime and k-1 divides n then T(n,k)=k, otherwise T(n,k)=1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 2, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 5, 1, 7, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gary W. Adamson & Mats Granvik, Aug 09 2008

By the von Stadt-Clausen theorem, the product of the terms in row n is the denominator of the Bernoulli number B_n.

			The triangle begins:
1,
1,2,
1,2,3,
1,1,1,1,
1,2,3,1,5,
1,1,1,1,1,1,
1,2,3,1,1,1,7,
1,1,1,1,1,1,1,1,
1,2,3,1,5,1,1,1,1,
1,1,1,1,1,1,1,1,1,1,
1,2,3,1,1,1,1,1,1,1,11,
1,1,1,1,1,1,1,1,1,1,1,1,
1,2,3,1,5,1,7,1,1,1,1,1,13,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,
...

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

Cf. A002445, A027642, A080092, A127093, A138239, A138243, A176079, A191904, A191910.

Entry revised by N. J. A. Sloane, Aug 10 2019

A176891 Triangle T(n,k) = k if k

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 4, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson and Mats Granvik, Apr 28 2010

A variant of A127093, which has T(n,n) = n. [The original definition said "Subsequence of A127093". Since all nonnegative integers are repeated infinitely often in both sequences, each one is a subsequence of the other, but there is no such relation on a row-by-row basis. - M. F. Hasler, Aug 08 2016]

Let A=A176891*A176891, B=A*A, C=B*B, D=C*C and so on, then the leftmost column in the last matrix (D) converges to A165552.

			Triangle begins:
1,
1,0,
1,0,0,
1,2,0,0,
1,0,0,0,0,
1,2,3,0,0,0,
1,0,0,0,0,0,0,
1,2,0,4,0,0,0,0,

Cf. A127093, A165552.

T(n,k) = if n=1 and k=1 then 1 elseif n=k then 0 elseif k divides n then k else 0.

Definition corrected by M. F. Hasler, Aug 08 2016

A127481 Triangle T(n,k) read by rows: T(n,k) = sum_{l=k..n, l|n, k|l} l*phi(k).

Original entry on oeis.org

1, 3, 2, 4, 0, 6, 7, 6, 0, 8, 6, 0, 0, 0, 20, 12, 8, 18, 0, 0, 12, 8, 0, 0, 0, 0, 0, 42, 15, 14, 0, 24, 0, 0, 0, 32, 13, 0, 24, 0, 0, 0, 0, 0, 54, 18, 12, 0, 0, 60, 0, 0, 0, 0, 40, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 28, 24, 42, 32, 0, 36, 0, 0, 0, 0, 0, 48, 14, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

			First few rows of the triangle are:
1;
3, 2;
4, 0, 6;
7, 6, 0, 8;
6, 0, 0, 0, 20,
12, 8, 18, 0, 0, 12;
8, 0, 0, 0, 0, 0, 42;
15, 14, 0, 24, 0, 0, 0, 32;
...

Cf. A054522, A127093, A001157 (row sums), A002618, A127466.

A127481 := proc(n,k)
    a :=0 ;
    for l from k to n do
        if modp(n,l) =0 and modp(l,k) =0 then
            a := a+l*numtheory[phi](k) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 06 2013

T(n,1) = A000203(n).
T(n,n) = A002618(n).
T(n,k) =sum_{l=k..n} A127093(n,l) * A054522(l,k), the matrix product of the infinite lower triangular matrices.

cp's OEIS Frontend

A340057 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A132442 Triangle whose n-th row consists of the first n terms of the n-th row of A134866.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A134866 Table read by antidiagonals: T(n,k) = sigma(gcd(n,k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A173541 Triangle read by rows: T(n,k)=k if k is a proper non-divisor of n, otherwise T(n,k)=0 (1<=k<=n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A228812 Triangle read by rows: T(n,k), n>=1, k>=1, in which row n lists m terms, where m = A055086(n). If k divides n and k < n^(1/2) then T(n,k) = k and T(n,m-k+1) = n/T(n,k). Also, if k^2 = n then T(n,k) = k. Other terms are zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs

A334947 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 6, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th octagonal number (A000567).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143112 A051731 * A032742 = sum of largest proper divisors of the divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A143343 Triangle T(n,k) (n>=0, 1<=k<=n+1) read by rows: T(n,1)=1 for n>=0, T(1,2)=2. If n>=3 is odd then T(n,k)=1 for all k. If n>=3 is even then if k is prime and k-1 divides n then T(n,k)=k, otherwise T(n,k)=1.

Views

Author

Keywords