A187208 -id:A187208 - cp's OEIS frontend

A187203 The bottom entry in the absolute difference triangle of the divisors of n.

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, 2, 8, 1, 16, 4, 18, 1, 8, 6, 22, 2, 16, 8, 8, 3, 28, 4, 30, 1, 8, 12, 24, 1, 36, 14, 8, 0, 40, 4, 42, 3, 20, 18, 46, 1, 36, 0, 8, 3, 52, 8, 36, 0, 8, 24, 58, 3, 60, 26, 4, 1, 40, 12, 66, 3, 8, 2, 70, 4, 72, 32, 32, 3

Offset: 1

Omar E. Pol, Aug 01 2011

nonn

Note that if n is prime then a(n) = n - 1.
Where records occurs gives the odd noncomposite numbers (A006005).
First differs from A187202 at a(14).
It is important to note that at each step in the process, the absolute differences are taken, and not just at the end. This sequence is therefore not abs(A187202) as I mistakenly assumed at first. - Alonso del Arte, Aug 01 2011

			a(18) = 4 because 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
with bottom entry a(18) = 4.
Note that A187202(18) = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A006005, A027750, A187202, A187205, A187208.

a187203 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = map abs $ zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

Table[d = Divisors[n]; While[Length[d] > 1, d = Abs[Differences[d]]]; d[[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)
Table[Nest[Abs[Differences[#]]&,Divisors[n],DivisorSigma[0,n]-1],{n,100}]//Flatten (* Harvey P. Dale, Nov 07 2022 *)

A187203(n)={ for(i=2,#n=divisors(n), n=abs(vecextract(n,"^1")-vecextract(n,"^-1"))); n[1]}  \\ M. F. Hasler, Aug 01 2011

Edited by Omar E. Pol, May 14 2016

A187207 Irregular triangle read by rows in which row n lists the k=A000005(n) divisors of n in decreasing order, followed by the lists of their absolute differences up to order k-1.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 2, 1, 2, 1, 1, 5, 1, 4, 6, 3, 2, 1, 3, 1, 1, 2, 0, 2, 7, 1, 6, 8, 4, 2, 1, 4, 2, 1, 2, 1, 1, 9, 3, 1, 6, 2, 4, 10, 5, 2, 1, 5, 3, 1, 2, 2, 0, 11, 1, 10, 12, 6, 4, 3, 2, 1, 6, 2, 1, 1, 1, 4, 1, 0, 0, 3, 1, 0, 2, 1, 1, 13, 1, 12, 14, 7, 2, 1, 7, 5, 1, 2, 4, 2

Offset: 1

Omar E. Pol, Aug 02 2011

			Triangle begins:
[1];
[2, 1], [1];
[3, 1], [2];
[4, 2, 1], [2, 1], [1];
[5, 1], [4];
[6, 3, 2, 1], [3, 1, 1], [2, 0], [2];
[7, 1], [6];
[8, 4, 2, 1], [4, 2, 1], [2, 1], [1];
[9, 3, 1], [6, 2], [4];
[10, 5, 2, 1], [5, 3, 1], [2, 2], [0];
The terms of each row can form a regular triangle, for example row 10:
10, 5, 2, 1;
. 5, 3, 1;
.   2, 2;
.    0;

Alois P. Heinz, Rows n = 1..350, flattened

Row n has length A184389(n) = A000217(A000005(n)). Row sums give A187215. Last terms of rows give A187203. Columns 1,2 give: A000027, A032742.

Cf. A056538, A187205, A187208.

with(numtheory):
DD:= l-> [seq(abs(l[i]-l[i-1]), i=2..nops(l))]:
T:= proc(n) local l;
      l:= sort([divisors(n)[]], `>`);
      seq((DD@@i)(l)[], i=0..nops(l)-1);
    end:
seq(T(n), n=1..20); # Alois P. Heinz, Aug 03 2011

row[n_] := (dd = Divisors[n]; Table[Differences[dd, k] // Reverse // Abs, {k, 0, Length[dd]-1}]); Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, May 18 2016 *)

A187209 Sum of all terms of triangle of absolute differences of the divisors of n.

Original entry on oeis.org

0, 1, 2, 4, 4, 9, 6, 11, 12, 13, 10, 24, 12, 21, 30, 26, 16, 43, 18, 40, 40, 37, 22, 59, 40, 45, 50, 52, 28, 89, 30, 57, 60, 61, 86, 90, 36, 69, 70, 103, 40, 125, 42, 88, 140, 85, 46, 128, 84, 97, 90, 106, 52, 165, 130, 113, 100, 109, 58, 201

Offset: 1

Omar E. Pol, Aug 04 2011

nonn

Note that if n is prime then a(n) = n - 1.

			a(10) = 13 because the divisors of 10 are 1, 2, 5, 10; the triangle of absolute differences is
1, 3, 5;
. 2, 2;
.   0;
and the sum of the terms of triangle is 1+3+5+2+2+0 = 13.

Cf. A187203, A187205, A187207, A187208.

Table[Total[Flatten[NestList[Abs[Differences[#]]&,Differences[Divisors[ n]], DivisorSigma[0,n]-1]]],{n,60}] (* Harvey P. Dale, Aug 10 2011 *)

a(n) = A187215(n) - A000203(n).

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A187203 The bottom entry in the absolute difference triangle of the divisors of n.

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A187207 Irregular triangle read by rows in which row n lists the k=A000005(n) divisors of n in decreasing order, followed by the lists of their absolute differences up to order k-1.

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A187209 Sum of all terms of triangle of absolute differences of the divisors of n.

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