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A273102
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Difference table of the divisors of the positive integers.
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17
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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
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OFFSET
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1,3
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COMMENTS
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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 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 A187202(n).
The sum of the elements of the slice n is A273103(n).
For another version see A273104, from which differs at a(92).
Each element of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)
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LINKS
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EXAMPLE
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For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, so the 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
-4 . 8
12
and the 18th slice is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2,-2, 6;
-4, 8;
12;
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 difference triangle of the divisors of n flattened. Row lengths are the terms of A184389. Row sums give A273103.
Triangle begins:
1;
1, 2, 1;
1, 3, 2;
1, 2, 4, 1, 2, 1;
...
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MATHEMATICA
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Table[Drop[FixedPointList[Differences, Divisors@ n], -2], {n, 15}] // Flatten (* Michael De Vlieger, May 16 2016 *)
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PROG
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(Sage)
def A273102_DTD(n): # DTD = Difference Table of Divisors
D = divisors(n)
T = matrix(ZZ, len(D))
for (m, d) in enumerate(D):
T[0, m] = d
for k in range(m-1, -1, -1) :
T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
return [T.row(k)[:len(D)-k] for k in range(len(D))]
# Keeps the rows of the DTD, for instance
# A273102_DTD(18)[1] = 1, 1, 3, 3, 9 (see the example above).
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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