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A299483 Irregular triangle read by rows in which row n lists the odd divisors of n in increasing order together with the even divisors of n in decreasing order.

%I A299483 #35 Jan 18 2019 04:42:45
%S A299483 1,1,2,1,3,1,4,2,1,5,1,3,6,2,1,7,1,8,4,2,1,3,9,1,5,10,2,1,11,1,3,12,6,
%T A299483 4,2,1,13,1,7,14,2,1,3,5,15,1,16,8,4,2,1,17,1,3,9,18,6,2,1,19,1,5,20,
%U A299483 10,4,2,1,3,7,21,1,11,22,2,1,23,1,3,24,12,8,6,4,2,1,5,25,1,13,26,2,1,3,9,27
%N A299483 Irregular triangle read by rows in which row n lists the odd divisors of n in increasing order together with the even divisors of n in decreasing order.
%C A299483 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 A299483 We can find the n-th row of the triangle as follows:
%C A299483 Consider only the semicircumferences that contain the point [n,0].
%C A299483 In the second quadrant from top to bottom we can see the curves that represent the odd divisors of n in increasing order. Also we can see these curves in the fourth quadrant from bottom to top.
%C A299483 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 A299483 Sequences of the same family are shown below:
%C A299483 -----------------------------------
%C A299483 Triangle    Order of divisors of n
%C A299483 -----------------------------------
%C A299483 A299481      odd  v   t.w.  even ^
%C A299483 This seq.    odd  ^   t.w.  even v
%C A299483 A319844      even v   t.w.  odd  ^
%C A299483 A319845      even ^   t.w.  odd  v
%C A299483 A319846      odd  v   t.w.  even v
%C A299483 A319847      odd  ^   t.w.  even ^
%C A299483 A319848      even v   t.w.  odd  v
%C A299483 A319849      even ^   t.w.  odd  ^
%C A299483 -----------------------------------
%C A299483 In the above table we have that:
%C A299483 "even v" means "even divisors of n in decreasing order".
%C A299483 "even ^" means "even divisors of n in increasing order".
%C A299483 "odd v"  means "odd divisors of n in decreasing order".
%C A299483 "odd ^"  means "odd divisors of n in increasing order".
%C A299483 "t.w." means "together with".
%H A299483 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 A299483 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A299483 Triangle begins:
%e A299483 1;
%e A299483 1,  2;
%e A299483 1,  3;
%e A299483 1,  4,  2;
%e A299483 1,  5;
%e A299483 1,  3,  6,  2;
%e A299483 1,  7;
%e A299483 1,  8,  4,  2;
%e A299483 1,  3,  9;
%e A299483 1,  5, 10,  2;
%e A299483 1, 11;
%e A299483 1,  3, 12,  6, 4, 2;
%e A299483 1, 13;
%e A299483 1,  7, 14,  2;
%e A299483 1,  3,  5, 15;
%e A299483 1, 16,  8,  4, 2;
%e A299483 1, 17;
%e A299483 1,  3,  9, 18, 6, 2;
%e A299483 1, 19;
%e A299483 1,  5, 20, 10, 4, 2;
%e A299483 1,  3,  7, 21;
%e A299483 1, 11, 22,  2;
%e A299483 1, 23;
%e A299483 1,  3, 24, 12, 8, 6, 4, 2;
%e A299483 1,  5, 25;
%e A299483 1, 13, 26,  2;
%e A299483 1,  3,  9, 27;
%e A299483 1,  7, 28, 14, 4, 2;
%e A299483 ...
%e A299483 For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. The odd divisors of 12 in increasing order are [1, 3], and the even divisors of 12 in decreasing order are [12, 6, 4, 2], so the 12th row of triangle is [1, 3, 12, 6, 4, 2].
%e A299483 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 top to bottom, we can see the curves with diameters [12, 4]. Also we can see these curves in the fourth quadrant from bottom to top. The associated numbers c = 12/d are [1, 3] respectively. These are the odd divisors of 12 in increasing 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 the 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 [1, 3, 12, 6, 4, 2] equaling the 12th row of triangle.
%o A299483 (PARI) row(n) = my(d=divisors(n)); concat(select(x->(x%2), d), Vecrev(select(x->!(x%2), d)));
%o A299483 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y A299483 Row sums give A000203.
%Y A299483 Row n has length A000005(n).
%Y A299483 Column 1 gives A000012.
%Y A299483 Right border gives A141310.
%Y A299483 Other permutations of A027750 are A056538, A210959, A299481, A319844, A319845, A319846, A319847, A319848, A319849.
%Y A299483 Cf. A001227, A183063, A299480, A299485.
%K A299483 nonn,tabf
%O A299483 1,3
%A A299483 _Omar E. Pol_, Feb 11 2018