A272210 -id:A272210 - cp's OEIS frontend

A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

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 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 antidiagonal sums give A273262.
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 A272121 at a(92).

			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 downwards gives the finite subsequence [1], [2, 1], [3, 1, 0], [6, 3, 2, 2], [9, 3, 0, -2, -4], [18, 9, 6, 6, 8, 12].

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A272210, A272121, A273102, A273103, A273262.

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

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

A273263 Irregular triangle read by rows: T(n,k) is the sum of the elements of the k-th column of the 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, 4, 9, 14, 14, 11, 13, 15, 15, 5, 8, 12, 16, 16, 17, 17, 12, 11, 12, 15, 18, 18, 19, 19, -3, 4, 10, 15, 20, 20, 13, 17, 21, 21, 4, 13, 22, 22, 23, 23, -4, 3, 8, 12, 16, 20, 24, 24, 21, 25, 25

Offset: 1

Omar E. Pol, May 22 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.
First differs from A274533 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;
   4,  9, 14, 14;
  11, 13, 15, 15;
   5,  8, 12, 16, 16;
  17, 17;
  12, 11, 12, 15, 18, 18;
  19, 19;
  -3,  4, 10, 15, 20, 20;
  13, 17, 21, 21;
   4, 13, 22, 22;
  23, 23;
  -4,  3,  8, 12, 16, 20, 24, 24;
  21, 25, 25;
   4, 15, 26, 26;
  ...
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 column sums give [12, 11, 12, 15, 18, 18] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Right border gives A000027. Column 1 is A161857. Row sums give A273103.

Cf. A187202, A273102, A273135, A272210, A273136, A273261, A273262, A274533.

Table[Total /@ Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 25}] // 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, j, sum(i=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 *)

cp's OEIS Frontend

A273135 Difference table of the divisors of the positive integers (with every table read by antidiagonals downwards).

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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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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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A273263 Irregular triangle read by rows: T(n,k) is the sum of the elements of the k-th column 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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