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A273262 Irregular triangle read by rows: T(n,k) = sum of the elements of the k-th antidiagonal of the difference table of the divisors of n. 5
1, 1, 3, 1, 5, 1, 3, 7, 1, 9, 1, 3, 4, 13, 1, 13, 1, 3, 7, 15, 1, 5, 19, 1, 3, 10, 17, 1, 21, 1, 3, 4, 5, 11, 28, 1, 25, 1, 3, 16, 21, 1, 5, 7, 41, 1, 3, 7, 15, 31, 1, 33, 1, 3, 4, 13, 6, 59, 1, 37, 1, 3, 7, 3, 31, 21, 1, 5, 13, 53, 1, 3, 28, 29, 1, 45, 1, 3, 4, 5, 11, 4, 36, 39, 1, 9, 61, 1, 3, 34, 33, 1, 5, 19, 65 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

If n is prime then row n contains only two terms: 1 and 2*n-1.

Row 2^k gives the first k+1 positive terms of A000225, k >= 0.

Note that this sequence contains negative terms.

First differs from A274532 at a(41).

LINKS

Table of n, a(n) for n=1..95.

EXAMPLE

Triangle begins:

1;

1, 3;

1, 5;

1, 3, 7;

1, 9;

1, 3, 4, 13;

1, 13;

1, 3, 7, 15;

1, 5, 19;

1, 3, 10, 17;

1, 21;

1, 3, 4, 5, 11, 28;

1, 25;

1, 3, 16, 21;

1, 5, 7, 41;

1, 3, 7, 15, 31;

1, 33;

1, 3, 4, 13, 6, 59;

1, 37;

1, 3, 7, 3, 31, 21;

1, 5, 13, 53;

1, 3, 28, 29;

1, 45;

1, 3, 4, 5, 11, 4, 36, 39;

1, 9, 61;

1, 3, 34, 33;

1, 5, 19, 65;

...

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 antidiagonal sums give [1, 3, 4, 13, 6, 59] which is also the 18th row of the irregular triangle.

MATHEMATICA

Table[Map[Total, Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}], {1}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 27}] (* 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, i, sum(k=0, i-1, m[i-k, k+1])); }

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. Column 1 is A000012. Right border gives A161700. Row sums give A273103.

Cf. A000225, A161700, A187202, A272210, A273102, A273135, A273261, A273263, A274532.

Sequence in context: A056753 A243158 A154723 * A274532 A254765 A300893

Adjacent sequences:  A273259 A273260 A273261 * A273263 A273264 A273265

KEYWORD

sign,tabf

AUTHOR

Omar E. Pol, May 20 2016

STATUS

approved

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Last modified June 23 09:36 EDT 2018. Contains 305693 sequences. (Running on oeis4.)