A273263 -id:A273263 - cp's OEIS frontend

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

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 *)

A273262 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the difference table of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 3, 7, 1, 9, 1, 3, 4, 13, 1, 13, 1, 3, 7, 15, 1, 5, 19, 1, 3, 10, 17, 1, 21, 1, 3, 4, 5, 11, 28, 1, 25, 1, 3, 16, 21, 1, 5, 7, 41, 1, 3, 7, 15, 31, 1, 33, 1, 3, 4, 13, 6, 59, 1, 37, 1, 3, 7, 3, 31, 21, 1, 5, 13, 53, 1, 3, 28, 29, 1, 45, 1, 3, 4, 5, 11, 4, 36, 39, 1, 9, 61, 1, 3, 34, 33, 1, 5, 19, 65

Offset: 1

Omar E. Pol, May 20 2016

If n is prime then row n contains only two terms: 1 and 2*n-1.
Row 2^k gives the first k+1 positive terms of A000225, k >= 0.
Note that this sequence contains negative terms.
First differs from A274532 at a(41).

			Triangle begins:
1;
1, 3;
1, 5;
1, 3, 7;
1, 9;
1, 3, 4, 13;
1, 13;
1, 3, 7, 15;
1, 5, 19;
1, 3, 10, 17;
1, 21;
1, 3, 4, 5, 11, 28;
1, 25;
1, 3, 16, 21;
1, 5, 7, 41;
1, 3, 7, 15, 31;
1, 33;
1, 3, 4, 13, 6, 59;
1, 37;
1, 3, 7, 3, 31, 21;
1, 5, 13, 53;
1, 3, 28, 29;
1, 45;
1, 3, 4, 5, 11, 4, 36, 39;
1, 9, 61;
1, 3, 34, 33;
1, 5, 19, 65;
...
For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, -2, 6;
-4, 8;
12;
The antidiagonal sums give [1, 3, 4, 13, 6, 59] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Column 1 is A000012. Right border gives A161700. Row sums give A273103.

Cf. A000225, A161700, A187202, A272210, A273102, A273135, A273261, A273263, A274532.

Table[Map[Total, Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}], {1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 27}] (* Michael De Vlieger, Jun 26 2016 *)

row(n) = {my(d = divisors(n)); my(nd = #d); my(m = matrix(#d, #d)); for (j=1, nd, m[1,j] = d[j];); for (i=2, nd, for (j=1, nd - i +1, m[i,j] = m[i-1,j+1] - m[i-1,j];);); vector(nd, i, sum(k=0, i-1, m[i-k, k+1]));}
tabf(nn) = for (n=1, nn, print(row(n)););
lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ Michel Marcus, Jun 25 2016

A273261 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the difference table of the divisors of n.

Original entry on oeis.org

1, 3, 1, 4, 2, 7, 3, 1, 6, 4, 12, 5, 2, 2, 8, 6, 15, 7, 3, 1, 13, 8, 4, 18, 9, 4, 0, 12, 10, 28, 11, 5, 4, 3, 1, 14, 12, 24, 13, 6, -2, 24, 14, 8, 8, 31, 15, 7, 3, 1, 18, 16, 39, 17, 8, 6, 4, 12, 20, 18, 42, 19, 9, 4, 3, -11, 32, 20, 12, 8, 36, 21, 10, -6, 24, 22, 60, 23, 11, 8, 6, 3, 4, -12, 31, 24, 16, 42, 25, 12, -8

Offset: 1

Omar E. Pol, May 20 2016

Row 2^k gives the first k+1 positive terms of A000225 in decreasing order, k >= 0.
If n is prime then row n contains only two terms: n+1 and n-1.
First differs from A274531 at a(41).
For n = p^k, T(n, 1) = n - 1, T(n, n) = (p - 1)^k. a(A006218(n - 1) + 1) = T(n, 0), a(A006218(n)) = T(n, t-1) where t is the number of divisors of n. - David A. Corneth, Jun 18 2016
Let D_n(m, c) be the k-th element in row m. The divisors of n are in row m = 0. Let t be the number of divisors of n. Then T(n, k) = D_n(k - 1, t-1) - D_n(k - 1, 0). - David A. Corneth, Jun 25 2016
For n in A187204, the last term of the n-th row is 0. - Michel Marcus, Apr 02 2017

			Triangle begins:
1;
3, 1;
4, 2;
7, 3, 1;
6, 4;
12, 5, 2, 2;
8, 6;
15, 7, 3, 1;
13, 8, 4;
18, 9, 4, 0;
12, 10;
28, 11, 5, 4, 3, 1;
14, 12;
24, 13, 6, -2;
24, 14, 8, 8;
31, 15, 7, 3, 1;
18, 16;
39, 17, 8, 6, 4, 12;
20, 18;
42, 19, 9, 4, 3, -11;
32, 20, 12, 8;
36, 21, 10, -6;
24, 22;
60, 23, 11, 8, 6, 3, 4, -12;
31, 24, 16;
42, 25, 12, -8;
...
For n = 14 the divisors of 14 are 1, 2, 7, 14, and the difference triangle of the divisors is
1, 2, 7, 14;
1, 5, 7;
4, 2;
-2;
The row sums give [24, 13, 6, -2] which is also the 14th row of the irregular triangle.
In the first row, the last element is 14, the first is 1. So the sum of the second row is 14 - 1 is 13. Similarly, the sum of the third row is 7 - 1 = 6, and of the last row, 2 - 4 = -2. - _David A. Corneth_, Jun 25 2016

Row lengths give A000005. Column 1 is A000203.
Right border gives A187202. Row sums give A273103.
Cf. A000225, A006218, A187204, A273102, A273262, A273263, A274531.

Map[Total, Table[NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 26}], {2}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)

row(n) = {my(d = divisors(n));my(nd = #d); my(m = matrix(#d, #d)); for (j=1, nd, m[1,j] = d[j];); for (i=2, nd, for (j=1, nd - i +1, m[i,j] = m[i-1,j+1] - m[i-1,j];);); vector(nd, i, sum(j=1, nd, m[i, j]));}
tabf(nn) = for (n=1, nn, print(row(n)););
lista(nn) = for (n=1, nn, v = row(n); for (j=1, #v, print1(v[j], ", "));); \\ Michel Marcus, Jun 25 2016

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 *)

A274533 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th column of the absolute difference table of the divisors of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 29 2016

If n is prime then row n is [n, n].
It appears that the last two terms of the n-th row are [n, n], n > 1.
Note that this sequence is not the absolute values of A273263.
First differs from A273263 at a(38).

			Triangle begins:
   1;
   2,  2;
   3,  3;
   3,  4,  4;
   5,  5;
   4,  5,  6,  6;
   7,  7;
   4,  6,  8,  8;
   7,  9,  9;
   4,  7, 10, 10;
  11, 11;
   4,  6,  8, 10, 12, 12;
  13, 13;
   8,  9, 14, 14;
  11, 13, 15, 15;
   5,  8, 12, 16, 16;
  17, 17;
   8, 11, 12, 15, 18, 18;
  19, 19;
   7, 10, 10, 15, 20, 20;
  13, 17, 21, 21;
  16, 13, 22, 22;
  23, 23;
   6,  7, 10, 12, 16, 20, 24, 24;
  21, 25, 25;
  20, 15, 26, 26;
  ...
For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the absolute difference triangle of the divisors is
  1,  2,  3,  6,  9, 18;
  1,  1,  3,  3,  9;
  0,  2,  0,  6;
  2,  2,  6;
  0,  4;
  4;
The column sums give [8, 11, 12, 15, 18, 18] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Right border gives A000027. Row sums give A187215.

Cf. A187203, A272121, A273132, A273104, A273137, A273263, A274531, A274532.

Table[Total /@ Table[#[[m - k + 1, -k]], {m, Length@ #, 1, -1}, {k, m}] &@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 25}] // Flatten (* Michael De Vlieger, Jun 29 2016 *)

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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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A273262 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the difference table of the divisors of n.

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A273261 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the difference table of the divisors of n.

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

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A274533 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th column of the absolute difference table of the divisors of n.

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