

A273262


Irregular triangle read by rows: T(n,k) = sum of the elements of the kth 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
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OFFSET

1,3


COMMENTS

If n is prime then row n contains only two terms: 1 and 2*n1.
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[i1, j+1]  m[i1, j]; ); ); vector(nd, i, sum(k=0, i1, m[ik, 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



