cp's OEIS Frontend

A319845 Irregular triangle read by rows in which row n lists the even divisors of n in increasing order together with the odd divisors of n in decreasing order.

%I A319845 #30 Jan 18 2019 04:41:29
%S A319845 1,2,1,3,1,2,4,1,5,1,2,6,3,1,7,1,2,4,8,1,9,3,1,2,10,5,1,11,1,2,4,6,12,
%T A319845 3,1,13,1,2,14,7,1,15,5,3,1,2,4,8,16,1,17,1,2,6,18,9,3,1,19,1,2,4,10,
%U A319845 20,5,1,21,7,3,1,2,22,11,1,23,1,2,4,6,8,12,24,3,1,25,5,1,2,26,13,1,27,9,3,1
%N A319845 Irregular triangle read by rows in which row n lists the even divisors of n in increasing order together with the odd divisors of n in decreasing order.
%C A319845 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 A319845 We can find the n-th row of the triangle as follows:
%C A319845 Consider only the semicircumferences that contain the point [n,0].
%C A319845 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.
%C A319845 Then, 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 (see example).
%C A319845 Sequences of the same family are shown below:
%C A319845 -----------------------------------
%C A319845 Triangle    Order of divisors of n
%C A319845 -----------------------------------
%C A319845 A299481      odd  v   t.w.  even ^
%C A319845 A299483      odd  ^   t.w.  even v
%C A319845 A319844      even v   t.w.  odd  ^
%C A319845 This seq.    even ^   t.w.  odd  v
%C A319845 A319846      odd  v   t.w.  even v
%C A319845 A319847      odd  ^   t.w.  even ^
%C A319845 A319848      even v   t.w.  odd  v
%C A319845 A319849      even ^   t.w.  odd  ^
%C A319845 -----------------------------------
%C A319845 In the above table we have that:
%C A319845 "even v" means "even divisors of n in decreasing order".
%C A319845 "even ^" means "even divisors of n in increasing order".
%C A319845 "odd v"  means "odd divisors of n in decreasing order".
%C A319845 "odd ^"  means "odd divisors of n in increasing order".
%C A319845 "t.w." means "together with".
%H A319845 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 A319845 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A319845 Triangle begins:
%e A319845    1;
%e A319845    2,  1;
%e A319845    3,  1;
%e A319845    2,  4,  1;
%e A319845    5,  1;
%e A319845    2,  6,  3,  1;
%e A319845    7,  1;
%e A319845    2,  4,  8,  1;
%e A319845    9,  3,  1;
%e A319845    2, 10,  5,  1;
%e A319845   11,  1;
%e A319845    2,  4,  6, 12,  3,  1;
%e A319845   13,  1;
%e A319845    2, 14,  7,  1;
%e A319845   15,  5,  3,  1;
%e A319845    2,  4,  8, 16,  1;
%e A319845   17,  1;
%e A319845    2,  6, 18,  9,  3,  1;
%e A319845   19,  1;
%e A319845    2,  4, 10, 20,  5,  1;
%e A319845   21,  7,  3,  1;
%e A319845    2, 22, 11,  1;
%e A319845   23,  1;
%e A319845    2,  4,  6,  8, 12, 24,  3,  1;
%e A319845   25,  5,  1;
%e A319845    2, 26, 13,  1;
%e A319845   27,  9,  3,  1;
%e A319845    2,  4, 14, 28,  7,  1;
%e A319845 ...
%e A319845 For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. The even divisors of 12 in increasing order are [2, 4, 6, 12], and the odd divisors of 12 in decreasing order are [3, 1], so the 12th row of triangle is [2, 4, 6, 12, 3, 1].
%e A319845 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 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. Then, 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. Finally all numbers c obtained are [2, 4, 6, 12, 3, 1] equaling the 12th row of triangle.
%o A319845 (PARI) row(n) = my(d=divisors(n)); concat(select(x->!(x%2), d), Vecrev(select(x->(x%2), d)));
%o A319845 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y A319845 Row sums give A000203.
%Y A319845 Row n has length A000005(n).
%Y A319845 Right border gives A000012.
%Y A319845 Other permutations of A027750 are A056538, A210959, A299481, A299483, A319844, A319846, A319847, A319848, A319849.
%Y A319845 Cf. A001227, A183063, A299480, A299485.
%K A319845 nonn,tabf
%O A319845 1,2
%A A319845 _Omar E. Pol_, Sep 29 2018