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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.
5
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
1,2
COMMENTS
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).
EXAMPLE
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.
MATHEMATICA
Table[Total /@ Transpose@ Map[Function[w, PadRight[w, Length@ #]], NestWhileList[Differences, #, Length@ # > 1 &]] &@ Divisors@ n, {n, 25}] // Flatten (* Michael De Vlieger, Jun 26 2016 *)
PROG
(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, 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
CROSSREFS
Row lengths give A000005. Right border gives A000027. Column 1 is A161857. Row sums give A273103.
Sequence in context: A189717 A219519 A330393 * A274533 A163127 A077113
KEYWORD
sign,tabf
AUTHOR
Omar E. Pol, May 22 2016
STATUS
approved