A244051 -id:A244051 - cp's OEIS frontend

A360461 T(n,k) is the sum of all the k-th smallest divisors of all positive integers <= n. Irregular triangle read by rows (n>=1, k>=1).

1, 2, 2, 3, 5, 4, 7, 4, 5, 12, 4, 6, 14, 7, 6, 7, 21, 7, 6, 8, 23, 11, 14, 9, 26, 20, 14, 10, 28, 25, 24, 11, 39, 25, 24, 12, 41, 28, 28, 6, 12, 13, 54, 28, 28, 6, 12, 14, 56, 35, 42, 6, 12, 15, 59, 40, 57, 6, 12, 16, 61, 44, 65, 22, 12, 17, 78, 44, 65, 22, 12, 18, 80, 47, 71, 31, 30, 19, 99, 47, 71, 31, 30

Offset: 1

Omar E. Pol, Feb 07 2023

Also, looking at all the partitions into equal-sized parts of all positive integers <= n, T(n,k) is the total number of parts in the partitions with the k-th largest parts.
Column k lists the partial sums of the column k of A027750.
The rows where the length row increases to a record gives A002182.

			Triangle begins:
   1;
   2,   2;
   3,   5;
   4,   7,  4;
   5,  12,  4;
   6,  14,  7,   6;
   7,  21,  7,   6;
   8,  23, 11,  14;
   9,  26, 20,  14;
  10,  28, 25,  24;
  11,  39, 25,  24;
  12,  41, 28,  28,  6, 12;
  ...
For n = 6 the divisors, in increasing order, of all positive integers <= 6 are as follows:
  -----------------------------
  n\k |    1     2     3     4
  -----------------------------
  1   |    1
  2   |    1,    2
  3   |    1,    3
  4   |    1,    2,    4
  5   |    1,    5
  6   |    1,    2,    3,    6
.
The sum of the first divisors (k = 1) is equal to 1+1+1+1+1+1 = 6, so T(6,1) = 6.
The sum of the second divisors (k = 2) is equal to 2+3+2+5+2 = 14, so T(6,2) = 14.
The sum of the third divisors (k = 3) is equal to 4+3 = 7, so T(6,3) = 7.
The sum of the fourth divisors (k = 4) is equal to 6, so T(6,4) = 6.
So the 6th row of the triangle is [6, 14, 7, 6].
Also, for n = 6 the partitions into equal parts, with the sizes of the parts in decreasing order, of all positive integers <= 6 are as follows:
  ----------------------------------------------------
  n\k |     1      2              3           4
  ----------------------------------------------------
  1   |    [1]
  2   |    [2],  [1,1]
  3   |    [3],  [1,1,1]
  4   |    [4],  [2,2],       [1,1,1,1]
  5   |    [5],  [1,1,1,1,1]
  6   |    [6],  [3,3],       [2,2,2],   [1,1,1,1,1,1]
.
The total number of parts in the 1st partitions (k = 1) is 6, so T(6,1) = 6.
The total number of parts in the 2nd partitions (k = 2) is 14, so T(6,2) = 14.
The total number of parts in the 3rd partitions (k = 3) is 7, so T(6,3) = 7.
The total number of parts in the 4th partitions (k = 4) is 6, so T(6,4) = 6.
So the 6th row of the triangle is [6, 14, 7, 6].

Paolo Xausa, Table of n, a(n) for n = 1..10175 (rows 1..550 of triangle, flattened)
Michael De Vlieger, Log log scatterplot of row n, n = 1..5040, showing T(n, k) with trajectory of k in a color function where black indicates k = 1, red k = 2, ..., magenta k = 60, ignoring zeros.

Row sums give A024916.
Row lengths give A070319.
Column 1 gives A000027.
Column 2 gives A088821.
The sum of the first n rows gives A175254.
Main sequences: A027750 and A244051.
Cf. A000005, A002182.

nn = 20; s[] = 0; m[0] = 0; Do[Set[m[n], Max[m[n - 1], DivisorSigma[0, n]]], {n, nn}]; Do[MapIndexed[(s[First[#2]] += #1; Set[t[n, First[#2]], s[First[#2]]]) &, PadRight[Divisors[n], m[n]]], {n, nn}]; Table[t[n, k], {n, nn}, {k, m[n]}] // Flatten (* _Michael De Vlieger, Mar 04 2023 *)
A360461[rowmax_]:=DeleteCases[Accumulate[PadRight[Divisors[Range[rowmax]]]],0,{2}];
A360461[20] (* Generates 20 rows *) (* Paolo Xausa, Mar 05 2023 *)

rowlen(n) = vecmax(vector(n, k, numdiv(k))); \\ A070319
row(n) = my(vd=vector(n, i, divisors(i)), nb=rowlen(n)); vector(nb, k, sum(i=1, #vd, if (#(vd[i]) >= k, vd[i][k]))); \\ Michel Marcus, Mar 06 2023

A354009 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of equal parts, in nonincreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 13 2022

The number of partitions in row n equals A001227(n), the number of odd divisors of n, and equals the number of partitions of n into consecutive parts, and equals the number of subparts in the symmetric representation of sigma(n).
The sum of row n equals A245579(n), the sum of all parts of all partitions of n into consecutive parts.
The length of row n equals A000593(n), the sum of the odd divisors of n.
Row n has length 1 if and only if n is a power of 2.
Is the right border the same as A006519?

			Triangle begins:
   [1];
   [2];
   [3], [1,1,1];
   [4];
   [5], [1,1,1,1,1];
   [6], [2,2,2];
   [7], [1,1,1,1,1,1,1];
   [8];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [2,2,2,2,2];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [4,4,4];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [2,2,2,2,2,2,2];
  [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [16];
  ...
For n = 10, in row 10 there are two partitions, equaling the number of odd divisors of 10, they are [1, 5], and equaling the number of partitions of 10 into consecutive parts, they are [10], [4, 3, 2, 1], and equaling the number of subparts in the symmetric representation of sigma(10), they are [9, 9].
The sum of row 10 is [10] + [2 + 2 + 2 + 2 + 2] = 20 equaling the sum of all parts of all partitions of 10 into consecutive parts, that is [10] + [4 + 3 + 2 + 1] = 20.
The length of row 10 is equal to 6 equaling the sum of the odd divisors of 10, that is 1 + 5 = 6.

Subsequence of A244051.
The number of partitions in row n equals A001227(n).
Row lengths give A000593.
Row sums give A245579.
Column 1 gives A000027.
Cf. A000079, A000203, A006519, A237593, A279387 (subparts), A299765.

Table[ConstantArray[n/#, #] & /@ Select[Divisors[n], OddQ], {n, 15}] // Flatten (* Michael De Vlieger, Jul 15 2022 *)

row(n) = my(v=[]); fordiv(n, d, if ((n/d)%2, v = concat(v, vector(n/d, k, d)))); Vecrev(v); \\ Michel Marcus, Jul 16 2022

cp's OEIS Frontend

A360461 T(n,k) is the sum of all the k-th smallest divisors of all positive integers <= n. Irregular triangle read by rows (n>=1, k>=1).

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