A187209 -id:A187209 - cp's OEIS frontend

A187215 Sum of the elements of the absolute difference table of the divisors of n.

1, 4, 6, 11, 10, 21, 14, 26, 25, 31, 22, 52, 26, 45, 54, 57, 34, 82, 38, 82, 72, 73, 46, 119, 71, 87, 90, 108, 58, 161, 62, 120, 108, 115, 134, 181, 74, 129, 126, 193, 82, 221, 86, 172, 218, 157, 94, 252, 141, 190, 162, 204, 106, 285, 202, 233

Offset: 1

Omar E. Pol, Aug 02 2011

First differs from A273103 at a(14). - Omar E. Pol, May 15 2016

			For n = 14 the divisors of 14 are 1, 2, 7, 14, and the absolute difference triangle of the divisors is
1 . 2 . 7 . 14
. 1 . 5 . 7
. . 4 . 2
. . . 2
The sum of all elements of the triangle is 1 + 2 + 7 + 14 + 1 + 5 + 7 + 4 + 2 + 2 = 45, so a(14) = 45.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Row sums of triangle A187207.

Cf. A000203, A056538, A187203, A187209, A273103.

with(numtheory):
DD:= l-> [seq(abs(l[i]-l[i-1]), i=2..nops(l))]:
a:= proc(n) local l;
      l:= sort([divisors(n)[]], `>`);
      add(j, j=[seq((DD@@i)(l)[], i=0..nops(l)-1)]);
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 02 2011

Table[Total@ Flatten@ NestWhileList[Abs@ Differences@ # &, Divisors@ n, Length@ # > 1 &], {n, 56}] (* Michael De Vlieger, May 18 2016 *)

a(n) = 2n, if n is prime.

a(2^k) = A125128(k+1), k >= 0. - Omar E. Pol, May 15 2016

More terms from Alois P. Heinz, Aug 02 2011

Edited by Omar E. Pol, May 19 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 *)

cp's OEIS Frontend

A187215 Sum of the elements of the absolute difference table of the divisors of n.

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