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A299481 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.

%I A299481 #61 Jan 18 2019 04:42:08
%S A299481 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 A299481 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 A299481 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 A299481 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 A299481 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 A299481 We can find the n-th row of the triangle as follows:
%C A299481 Consider only the semicircumferences that contain the point [n,0].
%C A299481 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 A299481 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 A299481 Sequences of the same family are shown below:
%C A299481 -----------------------------------
%C A299481 Triangle    Order of divisors of n
%C A299481 -----------------------------------
%C A299481 This seq.    odd  v   t.w.  even ^
%C A299481 A299483      odd  ^   t.w.  even v
%C A299481 A319844      even v   t.w.  odd  ^
%C A299481 A319845      even ^   t.w.  odd  v
%C A299481 A319846      odd  v   t.w.  even v
%C A299481 A319847      odd  ^   t.w.  even ^
%C A299481 A319848      even v   t.w.  odd  v
%C A299481 A319849      even ^   t.w.  odd  ^
%C A299481 -----------------------------------
%C A299481 In the above table we have that:
%C A299481 "even v" means "even divisors of n in decreasing order".
%C A299481 "even ^" means "even divisors of n in increasing order".
%C A299481 "odd v"  means "odd divisors of n in decreasing order".
%C A299481 "odd ^"  means "odd divisors of n in increasing order".
%C A299481 "t.w." means "together with".
%H A299481 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 A299481 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 A299481 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A299481 Triangle begins:
%e A299481    1;
%e A299481    1,  2;
%e A299481    3,  1;
%e A299481    1,  2,  4;
%e A299481    5,  1;
%e A299481    3,  1,  2,  6;
%e A299481    7,  1;
%e A299481    1,  2,  4,  8;
%e A299481    9,  3,  1;
%e A299481    5,  1,  2, 10;
%e A299481   11,  1;
%e A299481    3,  1,  2,  4,  6, 12;
%e A299481   13,  1;
%e A299481    7,  1,  2, 14;
%e A299481   15,  5,  3,  1;
%e A299481    1,  2,  4,  8, 16;
%e A299481   17,  1;
%e A299481    9,  3,  1,  2,  6, 18;
%e A299481   19,  1;
%e A299481    5,  1,  2,  4, 10, 20;
%e A299481   21,  7,  3,  1;
%e A299481   11,  1,  2, 22;
%e A299481   23,  1;
%e A299481    3,  1,  2,  4,  6,  8, 12, 24;
%e A299481   25,  5,  1;
%e A299481   13,  1,  2, 26;
%e A299481   27,  9,  3,  1;
%e A299481    7,  1,  2,  4, 14, 28;
%e A299481 ...
%e A299481 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 A299481 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 A299481 (PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->(x%2), d)), select(x->!(x%2), d));
%o A299481 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y A299481 Row sums give A000203.
%Y A299481 Row n has length A000005(n).
%Y A299481 Alternating borders give A000027.
%Y A299481 Right border gives A124625 without its first two terms.
%Y A299481 Other permutations of A027750 are A056538, A210959, A299483, A319844, A319845, A319846, A319847, A319848, A319849.
%Y A299481 Cf. A001227, A183063, A299480, A299485.
%K A299481 nonn,tabf
%O A299481 1,3
%A A299481 _Omar E. Pol_, Feb 10 2018