A299481 Irregular triangle read by rows in which row n lists the odd divisors of n in decreasing order together with the even divisors of n in increasing order.

1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 3, 1, 2, 6, 7, 1, 1, 2, 4, 8, 9, 3, 1, 5, 1, 2, 10, 11, 1, 3, 1, 2, 4, 6, 12, 13, 1, 7, 1, 2, 14, 15, 5, 3, 1, 1, 2, 4, 8, 16, 17, 1, 9, 3, 1, 2, 6, 18, 19, 1, 5, 1, 2, 4, 10, 20, 21, 7, 3, 1, 11, 1, 2, 22, 23, 1, 3, 1, 2, 4, 6, 8, 12, 24, 25, 5, 1, 13, 1, 2, 26, 27, 9, 3, 1

Offset: 1

Author

Omar E. Pol, Feb 10 2018

Keywords

• nonn
• tabf

Comments

Consider the diagram with overlapping periodic curves that appears in the Links section (figure 1). The number of curves that contain the point [n,0] equals the number of divisors of n. The simpler interpretation of the diagram is that the curve of diameter d represents the divisor d of n. Now here we introduce a new interpretation: the curve of diameter d that contains the point [n,0] represents the divisor c of n, where c = n/d. This version of the model has the property that each odd quadrant centered at [n,0] contains the curves that represent the even divisors of n, and each even quadrant centered at [n,0] contains the curves that represent the odd divisors of n.

We can find the n-th row of the triangle as follows:

Consider only the semicircumferences that contain the point [n,0].

In the second quadrant from bottom to top we can see the curves that represent the odd divisors of n in decreasing order. Also we can see these curves in the fourth quadrant from top to bottom.

Then, if n is an even number, in the first quadrant from top to bottom we can see the curves that represent the even divisors of n in increasing order. Also we can see these curves in the third quadrant from bottom to top (see example).

Sequences of the same family are shown below:

-----------------------------------

Triangle Order of divisors of n

-----------------------------------

This seq. odd v t.w. even ^

A299483 odd ^ t.w. even v

A319844 even v t.w. odd ^

A319845 even ^ t.w. odd v

A319846 odd v t.w. even v

A319847 odd ^ t.w. even ^

A319848 even v t.w. odd v

A319849 even ^ t.w. odd ^

-----------------------------------

In the above table we have that:

"even v" means "even divisors of n in decreasing order".

"even ^" means "even divisors of n in increasing order".

"odd v" means "odd divisors of n in decreasing order".

"odd ^" means "odd divisors of n in increasing order".

"t.w." means "together with".

Examples

			Triangle begins:
   1;
   1,  2;
   3,  1;
   1,  2,  4;
   5,  1;
   3,  1,  2,  6;
   7,  1;
   1,  2,  4,  8;
   9,  3,  1;
   5,  1,  2, 10;
  11,  1;
   3,  1,  2,  4,  6, 12;
  13,  1;
   7,  1,  2, 14;
  15,  5,  3,  1;
   1,  2,  4,  8, 16;
  17,  1;
   9,  3,  1,  2,  6, 18;
  19,  1;
   5,  1,  2,  4, 10, 20;
  21,  7,  3,  1;
  11,  1,  2, 22;
  23,  1;
   3,  1,  2,  4,  6,  8, 12, 24;
  25,  5,  1;
  13,  1,  2, 26;
  27,  9,  3,  1;
   7,  1,  2,  4, 14, 28;
...
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. The odd divisors of 12 in decreasing order are [3, 1], and the even divisors of 12 in increasing order are [2, 4, 6, 12], so the 12th row of triangle is [3, 1, 2, 4, 6, 12].
On the other hand, consider the diagram that appears in the Links section (figure 1). Then consider only the semicircumferences that contain the point [12,0]. In the second quadrant, from bottom to top, we can see the curves with diameters [4, 12]. Also we can see these curves in the fourth quadrant from top to bottom. The associated numbers c = 12/d are [3, 1] respectively. These are the odd divisors of 12 in decreasing order. Then, in the first quadrant, from top to bottom, we can see the curves with diameters [6, 3, 2, 1]. Also we can see these curves in the third quadrant from bottom to top. The associated numbers c = 12/d are [2, 4, 6, 12] respectively. These are the even divisors of n in increasing order. Finally all numbers c obtained are [3, 1, 2, 4, 6, 12] equaling the 12th row of triangle.
		

Links

• Omar E. Pol, Figure 1: Geometric model of divisors with periodic curves (for n = 1..16), figure 2: Upper part, figure 3: Lower part upside down.
• Omar E. Pol, Sobre el patrón de los números primos, and from Jason Davies, An interactive companion (for primes 2..997)
• Index entries for sequences related to divisors of numbers

Crossrefs

Row sums give A000203.

Row n has length A000005(n).

Alternating borders give A000027.

Right border gives A124625 without its first two terms.

Other permutations of A027750 are A056538, A210959, A299483, A319844, A319845, A319846, A319847, A319848, A319849.

Cf. A001227, A183063, A299480, A299485.

Programs

PARI

row(n) = my(d=divisors(n)); concat(Vecrev(select(x->(x%2), d)), select(x->!(x%2), d));
lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ Michel Marcus, Jan 17 2019