A184389 -id:A184389 - cp's OEIS frontend

A187207 Irregular triangle read by rows in which row n lists the k=A000005(n) divisors of n in decreasing order, followed by the lists of their absolute differences up to order k-1.

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

Offset: 1

Omar E. Pol, Aug 02 2011

			Triangle begins:
[1];
[2, 1], [1];
[3, 1], [2];
[4, 2, 1], [2, 1], [1];
[5, 1], [4];
[6, 3, 2, 1], [3, 1, 1], [2, 0], [2];
[7, 1], [6];
[8, 4, 2, 1], [4, 2, 1], [2, 1], [1];
[9, 3, 1], [6, 2], [4];
[10, 5, 2, 1], [5, 3, 1], [2, 2], [0];
The terms of each row can form a regular triangle, for example row 10:
10, 5, 2, 1;
. 5, 3, 1;
.   2, 2;
.    0;

Alois P. Heinz, Rows n = 1..350, flattened

Row n has length A184389(n) = A000217(A000005(n)). Row sums give A187215. Last terms of rows give A187203. Columns 1,2 give: A000027, A032742.

Cf. A056538, A187205, A187208.

with(numtheory):
DD:= l-> [seq(abs(l[i]-l[i-1]), i=2..nops(l))]:
T:= proc(n) local l;
      l:= sort([divisors(n)[]], `>`);
      seq((DD@@i)(l)[], i=0..nops(l)-1);
    end:
seq(T(n), n=1..20); # Alois P. Heinz, Aug 03 2011

row[n_] := (dd = Divisors[n]; Table[Differences[dd, k] // Reverse // Abs, {k, 0, Length[dd]-1}]); Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, May 18 2016 *)

A272121 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 18 2016

This is an irregular tetrahedron T(n,j,k) in which the slice n lists the elements of the j-th antidiagonal of the absolute difference triangle of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187203(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A187215(n).
If n is a power of 2 the antidiagonals are also the divisors of the powers of 2 from 1 to n in decreasing order, for example if n = 8 the finite sequence of antidiagonals is [1], [2, 1], [4, 2, 1], [8, 4, 2, 1].
First differs from A273135 at a(92).
Note that this sequence is not the absolute values of A273135. For example a(135) = 0 and A273135(135) = 4.

			The tables of the first nine positive integers are
  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;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
                 1;             0, 2;             1, 2;       4;
                                2;                1;
For n = 18 the absolute difference table of the divisors of 18 is
  1, 2, 3, 6, 9, 18;
  1, 1, 3, 3, 9;
  0, 2, 0, 6;
  2, 2, 6;
  0, 4;
  4;
This table read by antidiagonals downwards gives the finite subsequence [1], [2, 1], [3, 1, 0], [6, 3, 2, 2], [9, 3, 0, 2, 0], [18, 9, 6, 6, 4, 4].

Cf. A000005, A000217, A027750, A184389, A187203, A187215, A273104, A273132, A273135.

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m, 1, -1}] &@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

A272210 Difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 18 2016

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th antidiagonal (read upwards) of the difference table of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187202(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A273103(n).
The antidiagonal sums give A273262.
If n is a power of 2 the diagonals are also the divisors of the powers of 2 from 1 to n, for example if n = 8 the finite sequence of diagonals is [1], [1, 2], [1, 2, 4], [1, 2, 4, 8].
First differs from A273132 at a(89).

			The tables of the first nine positive integers are
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;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
.              1;             0, 2;             1, 2;       4;
.                             2;                1;
.
For n = 18 the difference table of the divisors of 18 is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, -2, 6;
-4, 8;
12;
This table read by antidiagonals upwards gives the finite subsequence [1], [1, 2], [0, 1, 3], [2, 2, 3, 6], [-4, -2, 0, 3, 9], [12, 8, 6, 6, 9, 18].

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A273102, A273103, A273109, A273135, A273132, A273136, A273261, A273262, A273263.

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 29 2016 *)

A273104 Absolute difference table of the divisors of the positive integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 15 2016

This is an irregular tetrahedron T(n,j,k) read by rows in which the slice n lists the elements of the rows of the absolute difference triangle of the divisors of n (including the divisors of n).
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187203(n).
The sum of the elements of the slice n is A187215(n).
For another version see A273102 from which differs at a(92).

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the absolute difference triangle of the divisors of 18 is
1 . 2 . 3 . 6 . 9 . 18
. 1 . 1 . 3 . 3 . 9
. . 0 . 2 . 0 . 6
. . . 2 . 2 . 6
. . . . 0 . 4
. . . . . 4
and the 18th slice is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, 2, 6;
0, 4;
4;
The tetrahedron begins:
1;
1, 2;
1;
1, 3;
2;
1, 2, 4;
1, 2;
1;
...
This is also an irregular triangle T(n,r) read by rows in which row n lists the absolute difference triangle of the divisors of n flattened.
Row lengths are the terms of A184389. Row sums give A187215.
Triangle begins:
1;
1, 2, 1;
1, 3, 2;
1, 2, 4, 1, 2, 1;
...

Cf. A027750, A184389, A187202-A187205, A187207-A187209, A187215, A273102.

Table[Drop[FixedPointList[Abs@ Differences@ # &, Divisors@ n], -2], {n, 15}] // Flatten (* Michael De Vlieger, May 16 2016 *)

A273132 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 18 2016

This is an irregular tetrahedron T(n,j,k) in which the slice n lists the elements of the j-th antidiagonal of the absolute difference triangle of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187203(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A187215(n).
If n is a power of 2 the antidiagonals are also the divisors of the powers of 2 from 1 to n, for example if n = 8 the finite sequence of antidiagonals is [1], [1, 2], [1, 2, 4], [1, 2, 4, 8].
First differs from A272210 at a(89).
Note that this sequence is not the absolute values of A272210. For example a(131) = 0 and A272210(131) = 4.

			The tables of the first nine positive integers are
  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;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
                 1;             0, 2;             1, 2;       4;
                                2;                1;
For n = 18 the absolute difference table of the divisors of 18 is
  1, 2, 3, 6, 9, 18;
  1, 1, 3, 3, 9;
  0, 2, 0, 6;
  2, 2, 6;
  0, 4;
  4;
This table read by antidiagonals upwards gives the finite subsequence [1], [1, 2], [0, 1, 3], [2, 2, 3, 6], [0, 2, 0, 3, 9], [4, 4, 6, 6, 9, 18].

Cf. A000005, A000217, A027750, A184389, A187203, A187215, A272121, A273104.

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}] &@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

A337297 a(n) = sigma(n)*(tau(n) - 1).

Original entry on oeis.org

0, 3, 4, 14, 6, 36, 8, 45, 26, 54, 12, 140, 14, 72, 72, 124, 18, 195, 20, 210, 96, 108, 24, 420, 62, 126, 120, 280, 30, 504, 32, 315, 144, 162, 144, 728, 38, 180, 168, 630, 42, 672, 44, 420, 390, 216, 48, 1116, 114, 465, 216, 490, 54, 840, 216, 840, 240, 270, 60, 1848, 62, 288

Offset: 1

Wesley Ivan Hurt, Aug 21 2020

Original name was: Sum of the coordinates of all pairs of divisors of n, (d1,d2), such that d1 < d2.

If n = p where p is prime, the only pair of divisors of p such that d1 < d2 is (1,p), whose coordinate sum is a(p) = p + 1. - Wesley Ivan Hurt, May 21 2021

			a(3) = 4; The divisors of 3 are {1,3}. If we form all ordered pairs (d1,d2) such that d1 < d2, we have: (1,3). The sum of the coordinates gives 1+3 = 4.
a(4) = 14; The divisors of 4 are {1,2,4}. If we form all ordered pairs (d1,d2) such that d1<d2, we have: (1,2), (1,4), (2,4). The sum of all the coordinates gives 1+2+1+4+2+4 = 14.

Cf. A066446, A184389, A337246.

Cf. A000005, A000203, A337360.

Table[Sum[Sum[(i + k)*(1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]

a(n) = my(d = divisors(n)); sum(i=1, #d, sum(j=1, i-1, d[i]+d[j])); \\ Michel Marcus, Aug 22 2020

a(n) = Sum_{d1|n, d2|n, d1

a(p^k) = k*(p^(k+1)-1)/(p-1) for p prime and k >= 1. - Wesley Ivan Hurt, Aug 23 2025

New name using formula from Ridouane Oudra, Jul 31 2025

A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 70, 105, 155, 223, 316, 443, 619, 865, 1210, 1690, 2354, 3263, 4497, 6157, 8368, 11280, 15078, 19989, 26296, 34356, 44626, 57693, 74321, 95503, 122535, 157101, 201377, 258155, 330994, 424398, 544035, 696995, 892104, 1140298, 1455080

Offset: 1

Gus Wiseman, Apr 30 2021

nonn

			The a(5) = 12 multisets of divisors:
  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
             {1,1,2}  {1,3}  {2}
             {1,2,2}  {3,3}  {4}
             {2,2,2}

Antidiagonal sums of the array A343658 (or row sums of the triangle).
Dominates A343657.
A000005 counts divisors.
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
A343656 counts divisors of powers.
Cf. A000169, A000312, A009998, A062319, A067824, A143773, A146291, A176029, A184389, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[Sum[multchoo[DivisorSigma[0,k],n-k],{k,n}],{n,10}]

a(n) = Sum_{k=1..n} binomial(sigma(k) + n - k - 1, n - k).

A273136 Difference table of the divisors of the positive integers (with every table read by columns).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 26 2016

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th column of the difference triangle of the divisors of n.
The first row of the slice n is also the n-th row of the triangle A027750.
The bottom entry of the slice n is A187202(n).
The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).
The sum of the elements of the n-th slice is A273103(n).
The columns sums give A273263.
If n is a power of 2 the subsequence lists the elements of the difference table of the divisors of n in nondecreasing order, for example if n = 8 the finite sequence of columns is [1, 1, 1, 1], [2, 2, 2], [4, 4], [8].
First differs from A273137 at a(86).

			The tables of the first nine positive integers are
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;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
.              1;             0, 2;             1, 2;       4;
.                             2;                1;
.
For n = 18 the difference table of the divisors of 18 is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, -2, 6;
-4, 8;
12;
This table read by columns gives the finite subsequence [1, 1, 0, 2, -4, 12], [2, 1, 2, -2, 8], [3, 3, 0, 6], [6, 3, 6], [9, 9], [18].

Cf. A000005, A000217, A027750, A184389, A187202, A272210, A273102, A273103, A273135, A273137, A273263.

Table[Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 15}] /. 0 -> {} // Flatten (* Michael De Vlieger, Jun 26 2016 *)

A337362 Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.

Original entry on oeis.org

1, 2, 3, 5, 3, 8, 3, 9, 6, 9, 3, 18, 3, 9, 10, 14, 3, 19, 3, 19, 10, 9, 3, 33, 6, 9, 10, 20, 3, 33, 3, 20, 10, 9, 10, 42, 3, 9, 10, 34, 3, 33, 3, 20, 21, 9, 3, 52, 6, 20, 10, 20, 3, 34, 10, 34, 10, 9, 3, 73, 3, 9, 21, 27, 10, 34, 3, 20, 10, 35, 3, 74, 3, 9, 21, 20, 10, 34, 3, 53, 15

Offset: 1

Wesley Ivan Hurt, Aug 24 2020

nonn

Number of distinct rectangles that can be made using the divisors of n as side lengths and whose length is never one more than its width.

			a(6) = 8; The divisors of 6 are {1,2,3,6}. There are 8 divisor pairs, (d1,d2), with d1 <= d2 that do not contain consecutive integers. They are (1,1), (1,3), (1,6), (2,2), (2,6), (3,3), (3,6) and (6,6). So a(6) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A337363.

Cf. also A335841, A337333, A184389, A129308.

Table[Sum[Sum[(1 - KroneckerDelta[i + 1, k]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

a(n) = sumdiv(n, d1, sumdiv(n, d2, (d1<=d2) && (d1 + 1 != d2))); \\ Michel Marcus, Aug 25 2020

a(n) = Sum_{d1|n, d2|n, d1<=d2} (1 - [d1 + 1 = d2]), where [] is the Iverson bracket.
a(n) = A337363(n) + A000005(n).
a(n) = A184389(n) - A129308(n). - Ridouane Oudra, Apr 15 2023

A160921 Numbers k such that k / (A000005(k)*(A000005(k)+1)/2) is an integer.

Original entry on oeis.org

1, 3, 10, 63, 147, 156, 225, 234, 408, 600, 680, 684, 952, 1014, 1496, 1500, 1768, 2176, 2584, 3128, 3944, 4216, 4224, 4275, 5032, 5576, 5848, 5880, 6392, 6498, 6660, 6875, 7208, 8024, 8296, 8379, 9112, 9324, 9656, 9840, 9928

Offset: 1

Ctibor O. Zizka, May 30 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..136 from R. J. Mathar)

Cf. A000005, A033950.

n :=1 :
for k from 1 to 50000 do
    if modp (k,A184389(k)) = 0 then
        printf("%d %d\n",n,k) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Oct 04 2014

t[n_] := n*(n + 1)/2; Select[Range[10^4], Divisible[#, t[DivisorSigma[0, #]]] &] (* Amiram Eldar, Jan 17 2021 *)
dsiQ[n_]:=With[{d=DivisorSigma[0,n]},IntegerQ[n/((d(d+1))/2)]]; Select[Range[10000],dsiQ] (* Harvey P. Dale, Aug 20 2023 *)

lista(nn) = {for (n=1, nn, if (2*n % (numdiv(n)*(numdiv(n)+1)) == 0, print1(n, ", ")););} \\ Michel Marcus, Jun 02 2013

Corrected by Michel Marcus, Jun 02 2013

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cp's OEIS Frontend

A187207 Irregular triangle read by rows in which row n lists the k=A000005(n) divisors of n in decreasing order, followed by the lists of their absolute differences up to order k-1.

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A272121 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

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A272210 Difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

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A273104 Absolute difference table of the divisors of the positive integers.

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A273132 Absolute difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

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A337297 a(n) = sigma(n)*(tau(n) - 1).

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PARI

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A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

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Formula

A273136 Difference table of the divisors of the positive integers (with every table read by columns).

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A337362 Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.

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