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.

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

Author

Omar E. Pol, May 20 2016

Keywords

• sign
• tabf

Comments

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

Examples

			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
		

Crossrefs

Row lengths give A000005. Column 1 is A000203.

Right border gives A187202. Row sums give A273103.

Cf. A000225, A006218, A187204, A273102, A273262, A273263, A274531.

Programs

Mathematica

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

PARI

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