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%I #61 Jan 18 2019 04:42:08
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C We can find the n-th row of the triangle as follows:
%C Consider only the semicircumferences that contain the point [n,0].
%C 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 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).
%C Sequences of the same family are shown below:
%C -----------------------------------
%C Triangle Order of divisors of n
%C -----------------------------------
%C This seq. odd v t.w. even ^
%C A299483 odd ^ t.w. even v
%C A319844 even v t.w. odd ^
%C A319845 even ^ t.w. odd v
%C A319846 odd v t.w. even v
%C A319847 odd ^ t.w. even ^
%C A319848 even v t.w. odd v
%C A319849 even ^ t.w. odd ^
%C -----------------------------------
%C In the above table we have that:
%C "even v" means "even divisors of n in decreasing order".
%C "even ^" means "even divisors of n in increasing order".
%C "odd v" means "odd divisors of n in decreasing order".
%C "odd ^" means "odd divisors of n in increasing order".
%C "t.w." means "together with".
%H 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 Omar E. Pol, <a href="http://www.polprimos.com">Sobre el patrón de los números primos</a>, and from Jason Davies, <a href="https://www.jasondavies.com/primos">An interactive companion (for primes 2..997)</a>
%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e Triangle begins:
%e 1;
%e 1, 2;
%e 3, 1;
%e 1, 2, 4;
%e 5, 1;
%e 3, 1, 2, 6;
%e 7, 1;
%e 1, 2, 4, 8;
%e 9, 3, 1;
%e 5, 1, 2, 10;
%e 11, 1;
%e 3, 1, 2, 4, 6, 12;
%e 13, 1;
%e 7, 1, 2, 14;
%e 15, 5, 3, 1;
%e 1, 2, 4, 8, 16;
%e 17, 1;
%e 9, 3, 1, 2, 6, 18;
%e 19, 1;
%e 5, 1, 2, 4, 10, 20;
%e 21, 7, 3, 1;
%e 11, 1, 2, 22;
%e 23, 1;
%e 3, 1, 2, 4, 6, 8, 12, 24;
%e 25, 5, 1;
%e 13, 1, 2, 26;
%e 27, 9, 3, 1;
%e 7, 1, 2, 4, 14, 28;
%e ...
%e 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].
%e 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.
%o (PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->(x%2), d)), select(x->!(x%2), d));
%o lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y Row sums give A000203.
%Y Row n has length A000005(n).
%Y Alternating borders give A000027.
%Y Right border gives A124625 without its first two terms.
%Y Other permutations of A027750 are A056538, A210959, A299483, A319844, A319845, A319846, A319847, A319848, A319849.
%Y Cf. A001227, A183063, A299480, A299485.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Feb 10 2018