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A319846 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 decreasing order. 7
1, 1, 2, 3, 1, 1, 4, 2, 5, 1, 3, 1, 6, 2, 7, 1, 1, 8, 4, 2, 9, 3, 1, 5, 1, 10, 2, 11, 1, 3, 1, 12, 6, 4, 2, 13, 1, 7, 1, 14, 2, 15, 5, 3, 1, 1, 16, 8, 4, 2, 17, 1, 9, 3, 1, 18, 6, 2, 19, 1, 5, 1, 20, 10, 4, 2, 21, 7, 3, 1, 11, 1, 22, 2, 23, 1, 3, 1, 24, 12, 8, 6, 4, 2, 25, 5, 1, 13, 1, 26, 2, 27, 9, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 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 bottom to top we can see the curves that represent the even divisors of n in decreasing order. Also we can see these curves in the third quadrant from top to bottom (see example).

Sequences of the same family are shown below:

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

Triangle    Order of divisors of n

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

A299481      odd  v   t.w.  even ^

A299483      odd  ^   t.w.  even v

A319844      even v   t.w.  odd  ^

A319845      even ^   t.w.  odd  v

This seq.    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".

LINKS

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

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.

Index entries for sequences related to divisors of numbers

EXAMPLE

Triangle begins:

   1;

   1,  2;

   3,  1;

   1,  4,  2;

   5,  1;

   3,  1,  6,  2;

   7,  1;

   1,  8,  4,  2;

   9,  3,  1;

   5,  1, 10,  2;

  11,  1;

   3,  1, 12,  6,  4,  2;

  13,  1;

   7,  1, 14,  2;

  15,  5,  3,  1;

   1, 16,  8,  4,  2;

  17,  1;

   9,  3,  1, 18,  6,  2;

  19,  1;

   5,  1, 20, 10,  4,  2;

  21,  7,  3,  1;

  11,  1, 22,  2;

  23,  1;

   3,  1, 24, 12,  8,  6,  4,  2;

  25,  5,  1;

  13,  1, 26,  2;

  27,  9,  3,  1;

   7,  1, 28, 14,  4,  2;

...

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 decreasing order are [12, 6, 4, 2], so the 12th row of triangle is [3, 1, 12, 6, 4, 2].

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 bottom to top, we can see the curves with diameters [1, 2, 3, 6]. Also we can see these curves in the third quadrant from top to bottom. The associated numbers c = 12/d are [12, 6, 4, 2] respectively. These are the even divisors of n in decreasing order. Finally all numbers c obtained are [3, 1, 12, 6, 4, 2] equaling the 12th row of triangle.

PROG

(PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->(x%2), d)), Vecrev(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

CROSSREFS

Row sums give A000203.

Row n has length A000005(n).

Other permutations of A027750 are A056538, A210959, A299481, A299483, A319844, A319845, A319847, A319848, A319849.

Cf. A001227, A183063, A299480, A299485.

Sequence in context: A023570 A256989 A275214 * A214690 A238878 A011249

Adjacent sequences:  A319843 A319844 A319845 * A319847 A319848 A319849

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Sep 29 2018

STATUS

approved

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Last modified June 30 09:10 EDT 2022. Contains 354920 sequences. (Running on oeis4.)