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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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=1..87.

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.

Cf. A187202, A273102, A273135, A272210, A273136, A273261, A273262, A274533.

Sequence in context: A189717 A219519 A330393 * A274533 A163127 A077113

Adjacent sequences:  A273260 A273261 A273262 * A273264 A273265 A273266

KEYWORD

sign,tabf

AUTHOR

Omar E. Pol, May 22 2016

STATUS

approved

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Last modified May 19 17:44 EDT 2022. Contains 353847 sequences. (Running on oeis4.)