A056538 -id:A056538 - cp's OEIS frontend

A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

1, 2, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 2, 3, 2, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 2, 2, 3, 1, 1, 6, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 1, 6, 3, 2, 1, 4, 1, 2, 1, 2, 2, 3, 1, 6, 3, 2, 4, 1, 1, 1, 1, 2, 2

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Also, largest number k such that, for each integer j from 1 to k, more multiples of j appear among the divisors of A002182(n) than appear among the divisors of any smaller positive integer.

For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k = n/j is highly composite (A002182).

			360 is a member of A002182, twice a member of A002182 (360/2 = 180), and three times a member of A002182 (360/3 = 120), but is not four times a member of A002182 (360/4 = 90 is not a member of A002182). Since A002182(13) = 360, a(13) = 3.
360 also sets records for the number of its divisors, the number of its divisors that are multiples of 2 (cf. A181808), and the number of its divisors that are multiples of 3, but not the number of its divisors that are multiples of 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Flammenkamp, List of the first 1200 highly composite numbers
Eric Weisstein's World of Mathematics, Highly composite number

a(n) equals the largest number k such that each number from 1 to k appears in row A002182(n) of A181803. a(n) also equals the largest number k such that each of the first k members of row A002182(n) of A056538 is highly composite.

A275055 Irregular triangle read by rows listing divisors d of n in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 4, 3, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 4, 8, 3, 6, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3

Offset: 1

Michael De Vlieger, Jul 14 2016

a(p^e) = A027750(p^e) for e >= 1.
The matrix of products that are divisors of n is arranged such that the powers of the prime divisors range across an axis, one axis per prime divisor. Thus a squarefree semiprime has a 2-dimensional matrix, a sphenic number has 3 dimensions, etc.
Generally, the number of dimensions for the matrix of divisors = omega(n) = A001221(n). Because of this, tau(n)*(mod omega(n)) = 0 for n > 1.
This follows from the formula for tau(n).
Prime divisors p of n are considered in numerical order.
Product matrix of tensors T = 1,p,p^2,...,p^e that include the powers 1 <= e of the prime divisor p that divide n.

			Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 5;
1, 2, 3, 6;
1, 7;
1, 2, 4, 8;
1, 3, 9;
1, 2, 5, 10;
1, 11;
1, 2, 4, 3, 6, 12;
1, 13;
1, 2, 7, 14;
1, 3, 5, 15;
1  2, 4, 8, 16;
1, 17;
1, 2, 3, 6, 9, 18;
...
2 prime divisors: n = 72
   1  2  4  8
   3  6 12 24
   9 18 36 72
   thus a(72) = {1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72}
3 prime divisors: n = 60
(the 3 dimensional levels correspond with powers of 5)
  level 5^0:        level 5^1:
   1  2  4    |     5  10  20
   3  6 12    |    15  30  60
   thus a(60) = {1, 2, 4, 3, 6, 12, 5, 10, 20, 15, 30, 60}
4 prime divisors: n = 210
(the 3 dimensional levels correspond with powers of 5,
the 4 dimensional levels correspond with powers of 7)
  level 5^0*7^0:    level 5^1*7^0:
     1   2     |     5  10
     3   6     |    15  30
  level 5^0*7^1:    level 5^1*7^1:
     7  14     |    35  70
    21  42     |   105 210
   thus a(210) = {1,2,3,6,5,10,15,30,7,14,21,42,35,70,105,210}

Michael De Vlieger, Table of n, a(n) for n = 1..11214 (Rows 1 <= n <= 1500)
Eric Weisstein's World of Mathematics, Divisor

Cf. A027750, A000005 (row length), A000203 (row sums), A056538.

{{1}}~Join~Table[TensorProduct @@ Reverse@ Apply[PowerRange[1, #1^#2, #1] &, # &@ FactorInteger@ n, 1], {n, 2, 30}] // Flatten

A225562 a(n) = smallest k such that n is the n-th largest divisor of k.

Original entry on oeis.org

1, 4, 15, 20, 30, 48, 84, 160, 144, 210, 462, 240, 624, 1134, 480, 864, 1836, 720, 8740, 840, 1512, 2376, 4968, 2400, 3900, 3120, 4536, 4032, 15312, 2520, 17856, 5280, 6930, 10710, 15400, 7200, 47952, 17100, 12480, 7920, 72324, 9240, 43344, 16632, 20790

Offset: 1

Irina Gerasimova, May 13 2013

nonn

The smallest row k such that n is the n-th entry in the triangle A056538 of divisors in reverse order.

Is a(n) defined for every n ? - Giovanni Resta, May 15 2013

			a(6) = 48 because the divisors of 48 are {48, 24, 16, 12, 8, 6, 4, 3, 2, 1} and 6 is the 6th divisor of 48.

Alois P. Heinz and Zak Seidov, Table of n, a(n) for n = 1..1000 (first 200 terms from Alois P. Heinz)

with(numtheory):
a:= proc(n) local k;
      for k from n by n while tau(k)`)[n]<>n do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 29 2013

a[n_] := Block[{k = 1, d}, While[Length[d = Reverse@ Divisors@ k] < n || n != d[[n]], k++]; k]; Array[a, 20] (* Giovanni Resta, May 15 2013 *)

a(13)-a(45) from Giovanni Resta, May 15 2013

A272024 Number of partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 3, 5, 15, 11, 77, 22, 176, 101, 385, 77, 3718, 135, 1575, 1575, 6842, 385, 31185, 627, 53174, 8349, 17977, 1575, 966467, 6842, 53174, 37338, 526823, 5604, 5392783, 8349, 1505499, 147273, 386155, 147273, 64112359, 26015, 966467, 526823, 56634173, 53174, 118114304, 75175, 26543660, 12132164, 5392783

Offset: 1

Omar E. Pol, Apr 19 2016

nonn

Also number of partitions of the total number of parts in the partitions of n into equal parts.

Note that one of the partitions of the sum of the divisors of n is also the list of divisors of n in decreasing order, see example.

			For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13 and the number of partitions of 13 is A000041(13) = 101, so a(9) = 101.
Note that one of the 101 partitions of 13 is [9, 3, 1] and it is also the list of divisors of 9 in decreasing order.

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

Cf. A000041, A000203, A056538, A058699, A072861, A139041, A272209.

Table[PartitionsP@ DivisorSigma[1, n], {n, 46}] (* Michael De Vlieger, Apr 19 2016 *)

a(n) = numbpart(sigma(n)); \\ Michel Marcus, Apr 19 2016

a(n) = p(sigma(n)) = A000041(A000203(n)).

A302170 Irregular triangle T(n,k) read by rows: first row is 1, n-th row (n > 1) lists distinct prime factors of n in decreasing order.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 2, 7, 2, 3, 5, 2, 11, 3, 2, 13, 7, 2, 5, 3, 2, 17, 3, 2, 19, 5, 2, 7, 3, 11, 2, 23, 3, 2, 5, 13, 2, 3, 7, 2, 29, 5, 3, 2, 31, 2, 11, 3, 17, 2, 7, 5, 3, 2, 37, 19, 2, 13, 3, 5, 2, 41, 7, 3, 2, 43, 11, 2, 5, 3, 23, 2, 47, 3, 2, 7, 5, 2, 17, 3, 13, 2, 53, 3, 2, 11, 5, 7, 2, 19, 3, 29, 2, 59, 5, 3, 2, 61, 31, 2

Offset: 1

Ilya Gutkovskiy, Apr 02 2018

			The irregular triangle begins:
1:  {1}
2:  {2}
3:  {3}
4:  {2}
5:  {5}
6:  {3, 2}
7:  {7}
8:  {2}
9:  {3}
10: {5, 2}
11: {11}
12: {3, 2}

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221 (row lengths), A006530, A008472 (row sums), A020639, A027746, A027748 (another version), A027750, A056538, A085307, A238689.

a302170 n k = a302170_tabl !! (n-1) !! (k-1)
a302170_tabl = map a302170_row [1..]
a302170_row = reverse . a027748_row
-- Brian Chess, Sep 19 2022

Flatten[Table[Reverse[FactorInteger[n][[All, 1]]], {n, 1, 62}]]

T(n,1) = A006530(n).

T(n,A001221(n)) = A020639(n).

A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d(m-1),m(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Jun 16 2025

Given a number d whose position is (n,k) so the next number d is in the position (n+d,k+d-1). In other words: we can find the next number d by walking d steps south and then walking d-1 steps east. Hence the Manhattan distance between two nearest numbers d is 2*d - 1.
By definition the m-th number d is in the position (d*(m-1),m*(d-1)) thus all the numbers d are on the same straight line.
The positive terms in the row n >= 1 are the divisors of n in increasing order. Hence the number of positive terms in row n is A000005(n).
The positive terms in the column k >= 1 are 1 plus the divisors of k in decreasing order. Hence the number of positive terms in the column k is A000005(k).
In the row n >= 1, two conjugate divisors of n are equidistant from the position (n,n-1). That position is in the same column that contains to the number n in the row 0.
In the column k >= 1, two conjugate (1 plus divisor)'s of k are equidistant from the position (k,k+1).
On the other hand we can find the divisors of n on a curve which starts at A(n-1,0) = 1 and ends at A(0,n-1) = n. Hence the number of positive terms in the curve (n-1) is A000005(n). Here that curve is called "curve of divisors of n".
Note that every divisor of n less to n on the curve is also a divisor of a number less to n in a row. Hence every divisor of n in a row is also a divisor on the curve of divisors of a number greater than n.
The position of two conjugate divisors of n on the curve is orthogonal and equidistant to the main diagonal of the array.
Curves never touch each other. See the curve and the row both with the divisors of 12 in the Example section.
Drawing all the curves the 2D structure is compatible with a 3D model where in orthogonal position is the arc diagram of A000005. See Links section.
The main diagonal is an irregular triangle read by rows in which row r lists r together with 2*r - 1 zeros, r >= 1.

			The corner 9 X 9 of the square array is as shown below:
.
   \k   0 1 2 3 4 5 6 7 8
   n\ _ _ _ _ _ _ _ _ _ _
     |
   0 |  1 2 3 4 5 6 7 8 9
   1 |  1 0 0 0 0 0 0 0 0
   2 |  1 0 2 0 0 0 0 0 0
   3 |  1 0 0 0 3 0 0 0 0
   4 |  1 0 0 2 0 0 4 0 0
   5 |  1 0 0 0 0 0 0 0 5
   6 |  1 0 0 0 2 0 3 0 0
   7 |  1 0 0 0 0 0 0 0 0
   8 |  1 0 0 0 0 2 0 0 0
  ...
.
The corner 25 X 25 of the square array without the zeros is as shown below:
.
                            1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
   \k   0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
   n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   0 |  1 2 3 4 5 6 7 8 9 ...
   1 |  1
   2 |  1   2
   3 |  1       3
   4 |  1     2     4
   5 |  1               5
   6 |  1       2   3       6
   7 |  1                       7
   8 |  1         2       4         8
   9 |  1               3               9
  10 |  1           2           5           10
  11 |  1                                       11
  12 |  1             2     3   4     6             12
  13 |  1                                               13
  14 |  1               2                   7
  15 |  1                       3       5
  16 |  1                 2           4           8
  17 |  1
  18 |  1                   2       3           6       9
  19 |  1
  20 |  1                     2             4   5
  21 |  1                               3               7
  22 |  1                       2
  23 |  1
  24 |  1                         2         3     4
  ...
.
The corner 25 X 25 of the square array without the zeros with the row and the curve of the divisors of 12 is as shown below:
.
                            1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
   \k   0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
   n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
     |
   0 |                        12
   1 |
   2 |
   3 |
   4 |
   5 |
   6 |                      6
   7 |
   8 |                    4
   9 |                  3
  10 |              2
  11 |  1
  12 |  1             2     3   4     6             12
  ...
.
The position of the conjugate divisors of 12 on the curve is symmetric respect to the main diagonal of the square array.
The position of the conjugate divisors of 12 in row 12 is symmetric respect the position (12,11). That position is in the same column that contains to the number 12 in the row 0.

Omar E. Pol, Illustration of initial terms of A000005 (arc diagram)

Row sums give A000203, n >= 1.
Column sums give A007503 = A000203 + A000005, k >= 1.
Row 0 gives A000027.
Column 0 gives A000012.
Cf. A027750, A056538, A208460.

A362310 Irregular triangle read by rows (row length A056220). Row n lists the integer solutions for x in the equation x - 10^n = x/y (x and y are integers).

Original entry on oeis.org

2, 5, 8, 9, 11, 12, 15, 20, 50, 75, 80, 90, 95, 96, 98, 99, 101, 102, 104, 105, 110, 120, 125, 150, 200, 500, 750, 800, 875, 900, 950, 960, 975, 980, 990, 992, 995, 996, 998, 999, 1001, 1002, 1004, 1005, 1008, 1010, 1020, 1025, 1040, 1050, 1100, 1125, 1200, 1250, 1500, 2000

Offset: 0

Thomas Scheuerle, Apr 15 2023

			Triangle begins:
  2
  5, 8, 9, 11, 12, 15, 20
  50, 75, 80, 90, 95, 96, 98, 99, 101, 102, 104, 105, 110, 120, 125, 150, 200
  ...
Corresponding values for y in the equation:
   2
  -1,-4,-9, 11, 6, 3, 2
  -1,-3,-4,-9,-19,-24,-49,-99, 101, 51, 26, 21, 11, 6, 5, 3, 2

Cf. A027750, A056220, A056538, A362311.

function a = A362310( max_row )
    r = 2; a= [];
    for n = 1:max_row
        d = divisors(10^(n-1));
        a = [a 10^(n-1)-d(end-1:-1:1) 10^(n-1)+d];
    end
end

row(n) = {my(d = divisors(10^n));my(v = concat(vector((n+1)^2,t,10^n)-Vecrev(d),vector((n+1)^2,t,10^n)+d));v[^1]};

Sum of row n = 2*(n+1)^2*10^n.
T(n, k) = 10^n - A056538(10^n, k+2), k < (n+1)^2-1.
T(n, k) = 10^n + A027750(10^n, k+2 - (n+1)^2), (n+1)^2-2 < k < 2*(n+1)^2-1.

A073083 Numbers n such that sum k/d(k) is an integer, where d(k) is the k-th divisor of n (the divisors of n are in decreasing order).

Original entry on oeis.org

1, 10, 12, 24, 615, 4066, 7960, 30432, 49260, 133686, 440286, 1201644, 6640812, 126953125, 411106256, 1046704882, 11046706752, 44588839041

Offset: 1

Benoit Cloitre, Aug 17 2002

a(12) > 11*10^5. - Emeric Deutsch, Mar 05 2005

8*10^11 < a(19) <= 2343594361433 = 13^10*17. It seems that the numbers 5^(13*k-3)*13 and 13^(17*k-7)*17, for k > 0, are terms. - Giovanni Resta, Dec 06 2019

			The divisors of 615 are [615,205,123,41,15,5,3,1] and 1/615+2/205+3/123+4/41+5/15+6/5+7/3+8/1 = 12 is an integer hence 615 is in the sequence.

Cf. A056538, A073082.

[k:k in [1..500000]|IsIntegral(&+[m/Reverse(Divisors(k))[m]:m in [1..#Divisors(k)]])]; // Marius A. Burtea, Dec 06 2019

Select[Range[441000],IntegerQ[Total[Range[DivisorSigma[0,#]]/ Reverse[ Divisors[ #]]]]&] (* Harvey P. Dale, May 23 2019 *)

More terms from Emeric Deutsch, Mar 05 2005
a(12)-a(17) from Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 15 2005
a(18) from Giovanni Resta, Dec 06 2019

A233772 Triangle read by rows in which row n lists n together with the proper divisors of n multiplied by -1 in increasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 03 2014

The same as A056538 but with all terms multiplied by -1 from the second column.

The sum of row n gives the deficiency of n.

			Written as an irregular triangle in which row n has length A000005(n) the sequence begins:
1;
2, -1;
3, -1;
4, -2, -1;
5, -1;
6, -3, -2, -1;
7, -1;
8, -4, -2, -1;
9, -3, -1;
10, -5, -2, -1;
11, -1;
12, -6, -4, -3, -2, -1;
...

Row sums give A033879.

Cf. A000005, A000203, A000396, A001065, A005100, A005101, A027750, A033880, A056538, A233773.

A235671 Triangle read by rows in which row n lists the proper divisors of n in increasing order, 2n, and the proper divisors of n in decreasing order.

Original entry on oeis.org

2, 1, 4, 1, 1, 6, 1, 1, 2, 8, 2, 1, 1, 10, 1, 1, 2, 3, 12, 3, 2, 1, 1, 14, 1, 1, 2, 4, 16, 4, 2, 1, 1, 3, 18, 3, 1, 1, 2, 5, 20, 5, 2, 1, 1, 22, 1, 1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1, 1, 26, 1, 1, 2, 7, 28, 7, 2, 1, 1, 3, 5, 30, 5, 3, 1, 1, 2, 4, 8, 32, 8, 4, 2, 1

Offset: 1

Omar E. Pol, Jan 24 2014

Numerators of a sequence related to the symmetric structure of sigma, which arises from the structure of A237593. The structure in the first two octants is transformed in a structure in the 6th and 7th octants, which is similar to an isosceles triangle.
Denominators are in A007395.
Row sums give A074400.
Row lengths is A114003 (see the Jovovic's formula in A114003).

			The irregular triangle begins:
2;
1, 4, 1;
1, 6, 1;
1, 2, 8, 2, 1;
1, 10, 1;
1, 2, 3, 12, 3, 2, 1;
1, 14, 1;
1, 2, 4, 16, 4, 2, 1;
1, 3, 18, 3, 1;
1, 2, 5, 20, 5, 2, 1;
1, 22, 1;
1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1;
...
Also:
1;
1/2, 2, 1/2;
1/2, 3, 1/2;
1/2, 1, 4, 1, 1/2;
1/2, 5, 1/2;
1/2, 1, 3/2, 6, 3/2, 1, 1/2;
1/2, 7, 1/2;
1/2, 1, 2, 8, 2, 1, 1/2;
1/2, 3/2, 9, 3/2, 1/2;
1/2, 1, 5/2, 10, 5/2, 1, 1/2;
1/2, 11, 1/2;
1/2, 1, 3/2, 2, 3, 12, 3, 2, 3/2, 1, 1/2;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A001065, A027750, A056538, A074400, A000203, A114002, A114003, A236104, A237591, A237593, A237270, A233772, A233773.

pd[n_]:=Module[{d=Most[Divisors[n]]},Flatten[Join[{d,{2n},Reverse[d]}]]]; Flatten[Array[pd,20]] (* Harvey P. Dale, Dec 22 2014 *)

cp's OEIS Frontend

A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

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A275055 Irregular triangle read by rows listing divisors d of n in order of appearance in a matrix of products that arranges the powers of prime divisors p of n along independent axes.

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A225562 a(n) = smallest k such that n is the n-th largest divisor of k.

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Maple

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A272024 Number of partitions of the sum of the divisors of n.

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A302170 Irregular triangle T(n,k) read by rows: first row is 1, n-th row (n > 1) lists distinct prime factors of n in decreasing order.

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Haskell

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A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d*(m-1),m*(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.

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A362310 Irregular triangle read by rows (row length A056220). Row n lists the integer solutions for x in the equation x - 10^n = x/y (x and y are integers).

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MATLAB

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A073083 Numbers n such that sum k/d(k) is an integer, where d(k) is the k-th divisor of n (the divisors of n are in decreasing order).

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A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d(m-1),m(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.