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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.

%I A319846 #30 Mar 10 2023 14:35:23
%S A319846 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,
%T A319846 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,
%U A319846 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
%N 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.
%C A319846 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.
%C A319846 We can find the n-th row of the triangle as follows:
%C A319846 Consider only the semicircumferences that contain the point [n,0].
%C A319846 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.
%C A319846 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).
%C A319846 Sequences of the same family are shown below:
%C A319846 -----------------------------------
%C A319846 Triangle    Order of divisors of n
%C A319846 -----------------------------------
%C A319846 A299481      odd  v   t.w.  even ^
%C A319846 A299483      odd  ^   t.w.  even v
%C A319846 A319844      even v   t.w.  odd  ^
%C A319846 A319845      even ^   t.w.  odd  v
%C A319846 This seq.    odd  v   t.w.  even v
%C A319846 A319847      odd  ^   t.w.  even ^
%C A319846 A319848      even v   t.w.  odd  v
%C A319846 A319849      even ^   t.w.  odd  ^
%C A319846 -----------------------------------
%C A319846 In the above table we have that:
%C A319846 "even v" means "even divisors of n in decreasing order".
%C A319846 "even ^" means "even divisors of n in increasing order".
%C A319846 "odd v"  means "odd divisors of n in decreasing order".
%C A319846 "odd ^"  means "odd divisors of n in increasing order".
%C A319846 "t.w." means "together with".
%H A319846 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv01.jpg">Figure 1: Geometric model of divisors with periodic curves (for n = 1..16)</a>, <a href="http://www.polprimos.com/imagenespub/poldiv02.jpg">figure 2: Upper part</a>, <a href="http://www.polprimos.com/imagenespub/poldiv03.jpg">figure 3: Lower part upside down.</a>
%H A319846 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A319846 Triangle begins:
%e A319846    1;
%e A319846    1,  2;
%e A319846    3,  1;
%e A319846    1,  4,  2;
%e A319846    5,  1;
%e A319846    3,  1,  6,  2;
%e A319846    7,  1;
%e A319846    1,  8,  4,  2;
%e A319846    9,  3,  1;
%e A319846    5,  1, 10,  2;
%e A319846   11,  1;
%e A319846    3,  1, 12,  6,  4,  2;
%e A319846   13,  1;
%e A319846    7,  1, 14,  2;
%e A319846   15,  5,  3,  1;
%e A319846    1, 16,  8,  4,  2;
%e A319846   17,  1;
%e A319846    9,  3,  1, 18,  6,  2;
%e A319846   19,  1;
%e A319846    5,  1, 20, 10,  4,  2;
%e A319846   21,  7,  3,  1;
%e A319846   11,  1, 22,  2;
%e A319846   23,  1;
%e A319846    3,  1, 24, 12,  8,  6,  4,  2;
%e A319846   25,  5,  1;
%e A319846   13,  1, 26,  2;
%e A319846   27,  9,  3,  1;
%e A319846    7,  1, 28, 14,  4,  2;
%e A319846 ...
%e A319846 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].
%e A319846 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.
%t A319846 Table[With[{d=Divisors[n]},Join[Reverse[Select[d,OddQ]],Reverse[Select[d,EvenQ]]]],{n,30}]//Flatten (* _Harvey P. Dale_, Mar 10 2023 *)
%o A319846 (PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->(x%2), d)), Vecrev(select(x->!(x%2), d)));
%o A319846 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y A319846 Row sums give A000203.
%Y A319846 Row n has length A000005(n).
%Y A319846 Other permutations of A027750 are A056538, A210959, A299481, A299483, A319844, A319845, A319847, A319848, A319849.
%Y A319846 Cf. A001227, A183063, A299480, A299485.
%K A319846 nonn,tabf
%O A319846 1,3
%A A319846 _Omar E. Pol_, Sep 29 2018