A274533 -id:A274533 - cp's OEIS frontend

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.

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

A274531 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th row of the absolute 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, 10, 4, 4, 20, 18, 42, 19, 11, 4, 5, 1, 32, 20, 12, 8, 36, 21, 10, 6, 24, 22, 60, 23, 11, 10, 6, 5, 2, 2, 31, 24, 16, 42, 25, 12, 8

Offset: 1

Omar E. Pol, Jun 27 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.
Note that this sequence is not the absolute values of A273261.
First differs from A273261 at a(41).

			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, 10, 4, 4;
20, 18;
42, 19, 11, 4, 5, 1;
32, 20, 12, 8;
36, 21, 10, 6;
24, 22;
60, 23, 11, 10, 6, 5, 2, 2;
31, 24, 16;
42, 25, 12, 8;
...
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 row sums give [39, 17, 8, 10, 4, 4] which is also the 18th row of the irregular triangle.

Row lengths give A000005. Column 1 is A000203. Right border gives A187203. Row sums give A187215.

Cf. A000225, A273104, A273261, A274532, A274533.

Map[Total, #, {2}] &@ Table[NestWhileList[Abs@ Differences@ # &, #, Length@ # > 1 &] &@ Divisors@ n, {n, 26}] // Flatten (* Michael De Vlieger, Jun 27 2016 *)

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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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