A127093 -id:A127093 - cp's OEIS frontend

A061150 a(n) = Sum_{d|n} d*prime(d).

2, 8, 17, 36, 57, 101, 121, 188, 224, 353, 343, 573, 535, 729, 777, 1036, 1005, 1406, 1275, 1801, 1669, 2087, 1911, 2861, 2482, 3167, 3005, 3753, 3163, 4541, 3939, 5228, 4879, 5737, 5391, 7314, 5811, 7475, 7063, 8873, 7341, 9957, 8215, 10607, 9849

Offset: 1

Vladeta Jovovic, Apr 16 2001

			a(4)=36 because the divisors of 4 are 1,2,4 and 1*p(1) + 2*p(2) + 4*p(4) = 1*2 + 2*3 + 4*7 = 36.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007441, A007445, A030009, A061151-A061152.

Cf. A127093.

with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(div[j]*ithprime(div[j]),j=1..tau(n)) end: seq(a(n),n=1..55); # Emeric Deutsch, Jan 20 2007

a(n) = sumdiv(n, d, d*prime(d)); \\ Michel Marcus, Jun 24 2018

Equals M * V, where M = A127093 as an infinite lower triangular matrix and V = A000040, the sequence of primes as a vector. E.g., a(4) = 36 = 1*2 + 2*3 + 4*7, where (1, 2, 0, 4) = row 4 of A127093 and 2, 3 and 7 are p(1), p(2), p(4). - Gary W. Adamson, Jan 11 2007

L.g.f.: log(Product_{k>=1} 1/(1 - x^k)^prime(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Edited by N. J. A. Sloane, May 04 2007

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 06 2006

An equivalent definition: Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127093 from the right. That is, T(n,m) = Sum_{j=m..n} A000012(n,j)*A127093(j,m) = Sum_{j=m..n} A127093(j,m) = m*floor(n/m) = m*A010766(n,m). - Gary W. Adamson, Jan 05 2007

The number of parts k in the triangle is A000203(k) hence the sum of parts k is A064987(k). - Omar E. Pol, Jul 05 2014

			Triangle begins:
{1},
{2, 2},
{3, 2, 3},
{4, 4, 3, 4},
{5, 4, 3, 4, 5},
{6, 6, 6, 4, 5, 6},
{7, 6, 6, 4, 5, 6, 7},
{8, 8, 6, 8, 5, 6, 7, 8},
{9, 8, 9, 8, 5, 6, 7, 8, 9},
...

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

Cf. A074025, A093960.

Cf. also A024916 (row sums), A127093, A127094, A127096, A127097, A127098, A127099, A038040, A000203, A126988, A127013, A127057.

Flat(List([1..10],n->List([1..n],m->n-(n mod m)))); # Muniru A Asiru, Oct 12 2018

seq(seq(n-modp(n,m),m=1..n),n=1..13); # Muniru A Asiru, Oct 12 2018

a = Table[Table[n - Mod[n, m], {m, 1, n}], {n, 1, 20}]; Flatten[a]

for(n=1,9,for(m=1,n,print1(n-n%m", "))) \\ Charles R Greathouse IV, Nov 07 2011

Edited by N. J. A. Sloane, Jul 05 2014 at the suggestion of Omar E. Pol, who observed that A127095 (Gary W. Adamson, with edits by R. J. Mathar) was the same as this sequence.

A330466 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 2, n >= 1, k >= 1, and the first element of column k is in row k^2.

Original entry on oeis.org

1, 1, 1, 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, 4, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 0, 1, 2, 0, 4, 1, 0, 3, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 2, 3, 4, 1, 0, 0, 0, 5, 1, 2, 0, 0, 0, 1, 0, 3, 0, 0, 1, 2, 0, 4, 0, 1, 0, 0, 0, 0, 1, 2, 3, 0, 5, 1, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Apr 30 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 row k^2.
Conjecture: row sums give A066839.

			Triangle begins (rows 1..25):
1;
1;
1;
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, 4;
1, 0, 0, 0;
1, 2, 3, 0;
1, 0, 0, 0;
1, 2, 0, 4;
1, 0, 3, 0;
1, 2, 0, 0;
1, 0, 0, 0;
1, 2, 3, 4;
1, 0, 0, 0, 5;
...
For n = 16 there are three partitions of 16 into consecutive parts that differ by 2, including 16 as a partition. They are [16], [9, 7] and [7, 5, 3, 1]. The number of parts of these partitions are 1, 2 and 4 respectively, so the 16th row of the triangle is [1, 2, 0, 4].

Cf. A000290, A066839, A237048, A303300.

Other triangles of the same family are A127093 and A285914.

T(n,k) = k*A303300(n,k).

A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 4, 1, 3, 2, 4, 3, 1, 5, 1, 2, 4, 2, 6, 3, 6, 5, 1, 2, 3, 6, 1, 5, 2, 4, 8, 3, 9, 5, 10, 7, 1, 7, 1, 2, 3, 6, 2, 10, 3, 6, 12, 5, 15, 7, 14, 11, 1, 2, 4, 8, 1, 7, 2, 4, 6, 12, 3, 15, 5, 10, 20, 7, 21, 11, 22, 15, 1, 3, 9, 1, 2, 4, 8, 2, 14, 3, 6, 9, 18, 5

Offset: 1

Omar E. Pol, Dec 27 2020

This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
  [1];
  [1, 2],    [1];
  [1, 3],    [1, 2],    [2];
  [1, 2, 4], [1, 3],    [2, 4], [3];
  [1, 5],    [1, 2, 4], [2, 6], [3, 6], [5];
  [...
The row sums of triangle give A066186.
Written as an irregular tetrahedron the first five slices are:
  1;
  -----
  1, 2,
  1;
  -----
  1, 3,
  1, 2,
  2;
  --------
  1, 2, 4,
  1, 3,
  2, 4,
  3;
  --------
  1, 5,
  1, 2, 4,
  2, 6,
  3, 6,
  5;
  --------
The row sums of tetrahedron give A339106.
The slices of the tetrahedron appear in the following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A027750 |     |       |  1      |  1 2      |  1   3      |
| I | A027750 |     |       |  1      |  1 2      |  1   3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O | A027750 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
| C | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
| O |    -    |     |       |  2      |  2 4      |  2   6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The lower zone is a condensed version of the "divisors" zone.

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

Nonzero terms of A340011.
Row sums give A066186.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340061.

A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m],{m,0,n-1}]];Array[A340056row,10] (* Paolo Xausa, Sep 01 2023 *)

A134577 A127170 * A127648.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 02 2007

Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, 26, ...).
Left border = A000005: (1, 2, 2, 3, 2, 4, 2, ...).
A134577 * [1/1, 1/2, 1/3, ...] = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).
A134577 * [1, 2, 3, ...] = A007433: (1, 6, 11, 27, 27, 66, ...).
A134577 * A000005 = A034761: (1, 6, 8, 23, 12, 48, ...).

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

Cf. A051731, A127170, A127648, A127093, A007429, A127170, A007433, A034761.

A127170 * A127648 = A051731^2 * A127648 = A051731 * A127093.

A334462 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 4, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 05 2020

Since the trivial partition n is counted, so T(n,1) = 1.
This is also 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 hexagonal 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..28):
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, 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;
...
For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. The number of parts of these partitions are 1, 2, 4 respectively, so the 28th row of the triangle is [1, 2, 0, 4].

Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), this sequence (d=4), A334540 (d=5).

Cf. A000384, A334460, A334461.

T(n,k) = k*A334460(n,k).

A127096 Triangle T(n,m) = A000012*A127094 read by rows.

Original entry on oeis.org

1, 3, 1, 6, 1, 1, 10, 1, 3, 1, 15, 1, 3, 1, 1, 21, 1, 3, 4, 3, 1, 28, 1, 3, 4, 3, 1, 1, 36, 1, 3, 4, 7, 1, 3, 1, 45, 1, 3, 4, 7, 1, 6, 1, 1, 55, 1, 3, 4, 7, 6, 6, 1, 3, 1, 66, 1, 3, 4, 7, 6, 6, 1, 3, 1, 1, 78, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 91, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 1, 105, 1, 3, 4, 7, 6, 12, 8, 7, 4, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Jan 05 2007

Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127094 from the right.

			First few rows of the triangle are:
   1;
   3, 1,
   6, 1, 1;
  10, 1, 3, 1;
  15, 1, 3, 1, 1;
  21, 1, 3, 4, 3, 1;
  28, 1, 3, 4, 3, 1, 1;
  ...

Cf. A127093, A127094, A123229, A024916 (row sums), A000203, A126988.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A127094 := proc(n,m) A127093(n, n-m+1) ; end:
A127096 := proc(n,m) add( A127094(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127096(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T[n_, m_] := Sum[1 + Mod[j, m - j - 1] - Mod[1 + j, m - j - 1], {j, m, n}];
Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2023 *)

T(n,m) = Sum_{j=m..n} A000012(n,j)*A127094(j,m) = Sum_{j=m..n} A127094(j,m).

Edited and extended by R. J. Mathar, Aug 18 2009

A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 0, 6, 4, 4, 0, 8, 5, 0, 0, 0, 20, 6, 6, 12, 0, 0, 12, 7, 0, 0, 0, 0, 0, 42, 8, 8, 0, 16, 0, 0, 0, 32, 9, 0, 18, 0, 0, 0, 0, 0, 54, 10, 10, 0, 0, 40, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 15 2007

Mobius transform of A127481.

			First few rows of the triangle are:
1;
2, 2;
3, 0, 6;
4, 4, 0, 8;
5, 0, 0, 0, 20;
6, 6, 12, 0, 0, 12;
7, 0, 0, 0, 0, 0, 42;
8, 8, 0, 16, 0, 0, 0, 32;
...

Cf. A054522, A127093, A062952, A002618.

Cf. A127093, A127481, A054525.

A127466 := proc(n,k)
    add( A054525(n,j)*A127481(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 06 2013

Sum_{k=1..n} T(n,k) = n^2.

T(n,n) = A002618(n) = n*phi(n).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A191904 Square array read by antidiagonals up: T(n,k) = 1-k if k divides n, else 1.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Jun 19 2011

Transposed variant of A177121. Array variant of A176079.

			Table begins:
0..1..1..1..1..1..1..1..1...
0.-1..1..1..1..1..1..1..1...
0..1.-2..1..1..1..1..1..1...
0.-1..1.-3..1..1..1..1..1...
0..1..1..1.-4..1..1..1..1...
0.-1.-2..1..1.-5..1..1..1...
0..1..1..1..1..1.-6..1..1...
0.-1..1.-3..1..1..1.-7..1...
0..1.-2..1..1..1..1..1.-8...

Mats Granvik, Mathematica program for recurrences.

Cf. A177121, A127093, A175992, A191907, A156734.

Cf. A176079.

nn = 30; t[n_, k_] := t[n, k] = If[Mod[n, k] == 0, -(k - 1), 1]; MatrixForm[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]

Conjecture: Sum_{n>=1} T(n,k)/n = log(k).
From Mats Granvik, Apr 24 2022: (Start)
Sum recurrence:
T(n, 1) = [n >= 1]*0;
T(n, k) = [n < k]*1;
T(n, k) = [n >= k](Sum_{i=1..k-1} T(n - i, k - 1) - Sum_{i=1..k-1} T(n - i, k)).
Product recurrence:
T(n, 1) = [n >= 1]*0;
T(n, k) = [n < k]*1;
T(n, k) = [n >= k](Product_{i=1..k-1} T(n - i, k - 1) - Product_{i=1..k-1} T(n - i, k)).
(End)

A334540 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 5, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th heptagonal number (A000566).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 05 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 heptagonal 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..27):
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, 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;
...
For n = 27 there are three partitions of 27 into consecutive parts that differ by 5, including 27 as a valid partition. They are [27], [16, 11] and [14, 9, 4]. The number of parts of these partitions are 1, 2, 3 respectively, so the 27th row of the triangle is [1, 2, 3].

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), this sequence (d=5).

Cf. A000566, A334465.

A334540 := proc(n,k)
        k*A334465(n,k) ;
end proc: # R. J. Mathar, Oct 02 2020

T(n,k) = k*A334465(n,k).

cp's OEIS Frontend

A061150 a(n) = Sum_{d|n} d*prime(d).

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A123229 Triangle read by rows: T(n, m) = n - (n mod m).

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A330466 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 2, n >= 1, k >= 1, and the first element of column k is in row k^2.

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A340056 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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A134577 A127170 * A127648.

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A334462 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 4, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

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A127096 Triangle T(n,m) = A000012*A127094 read by rows.

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A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

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