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:
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Triangle Order of divisors of n
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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 ^
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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
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.
MATHEMATICA
Table[With[{d=Divisors[n]}, Join[Reverse[Select[d, OddQ]], Reverse[Select[d, EvenQ]]]], {n, 30}]//Flatten (* Harvey P. Dale, Mar 10 2023 *)
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
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Sep 29 2018
STATUS
approved