A235671 Triangle read by rows in which row n lists the proper divisors of n in increasing order, 2n, and the proper divisors of n in decreasing order.

2, 1, 4, 1, 1, 6, 1, 1, 2, 8, 2, 1, 1, 10, 1, 1, 2, 3, 12, 3, 2, 1, 1, 14, 1, 1, 2, 4, 16, 4, 2, 1, 1, 3, 18, 3, 1, 1, 2, 5, 20, 5, 2, 1, 1, 22, 1, 1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1, 1, 26, 1, 1, 2, 7, 28, 7, 2, 1, 1, 3, 5, 30, 5, 3, 1, 1, 2, 4, 8, 32, 8, 4, 2, 1

Offset

1,1

Comments

Numerators of a sequence related to the symmetric structure of sigma, which arises from the structure of A237593. The structure in the first two octants is transformed in a structure in the 6th and 7th octants, which is similar to an isosceles triangle.

Denominators are in A007395.

Row sums give A074400.

Row lengths is A114003 (see the Jovovic's formula in A114003).

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

The irregular triangle begins:
2;
1, 4, 1;
1, 6, 1;
1, 2, 8, 2, 1;
1, 10, 1;
1, 2, 3, 12, 3, 2, 1;
1, 14, 1;
1, 2, 4, 16, 4, 2, 1;
1, 3, 18, 3, 1;
1, 2, 5, 20, 5, 2, 1;
1, 22, 1;
1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1;
...
Also:
1;
1/2, 2, 1/2;
1/2, 3, 1/2;
1/2, 1, 4, 1, 1/2;
1/2, 5, 1/2;
1/2, 1, 3/2, 6, 3/2, 1, 1/2;
1/2, 7, 1/2;
1/2, 1, 2, 8, 2, 1, 1/2;
1/2, 3/2, 9, 3/2, 1/2;
1/2, 1, 5/2, 10, 5/2, 1, 1/2;
1/2, 11, 1/2;
1/2, 1, 3/2, 2, 3, 12, 3, 2, 3/2, 1, 1/2;
...

Mathematica

pd[n_]:=Module[{d=Most[Divisors[n]]}, Flatten[Join[{d, {2n}, Reverse[d]}]]]; Flatten[Array[pd, 20]] (* Harvey P. Dale, Dec 22 2014 *)

Crossrefs

Cf. A000005, A001065, A027750, A056538, A074400, A000203, A114002, A114003, A236104, A237591, A237593, A237270, A233772, A233773.

Sequence in context: A131035 A339833 A118745 * A375495 A131034 A346873

Adjacent sequences: A235668 A235669 A235670 * A235672 A235673 A235674

Keyword

nonn,frac,tabf

Author

Omar E. Pol, Jan 24 2014

Status

approved