cp's OEIS Frontend

A319844 Irregular triangle read by rows in which row n lists the even divisors of n in decreasing order together with the odd divisors of n in increasing order.

%I A319844 #33 Jan 18 2019 04:43:03
%S A319844 1,2,1,1,3,4,2,1,1,5,6,2,1,3,1,7,8,4,2,1,1,3,9,10,2,1,5,1,11,12,6,4,2,
%T A319844 1,3,1,13,14,2,1,7,1,3,5,15,16,8,4,2,1,1,17,18,6,2,1,3,9,1,19,20,10,4,
%U A319844 2,1,5,1,3,7,21,22,2,1,11,1,23,24,12,8,6,4,2,1,3,1,5,25,26,2,1,13,1,3,9,27
%N A319844 Irregular triangle read by rows in which row n lists the even divisors of n in decreasing order together with the odd divisors of n in increasing order.
%C A319844 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 A319844 We can find the n-th row of the triangle as follows:
%C A319844 Consider only the semicircumferences that contain the point [n,0].
%C A319844 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.
%C A319844 Then, 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 (see example).
%C A319844 Sequences of the same family are shown below:
%C A319844 -----------------------------------
%C A319844 Triangle    Order of divisors of n
%C A319844 -----------------------------------
%C A319844 A299481      odd  v   t.w.  even ^
%C A319844 A299483      odd  ^   t.w.  even v
%C A319844 This seq.    even v   t.w.  odd  ^
%C A319844 A319845      even ^   t.w.  odd  v
%C A319844 A319846      odd  v   t.w.  even v
%C A319844 A319847      odd  ^   t.w.  even ^
%C A319844 A319848      even v   t.w.  odd  v
%C A319844 A319849      even ^   t.w.  odd  ^
%C A319844 -----------------------------------
%C A319844 In the above table we have that:
%C A319844 "even v" means "even divisors of n in decreasing order".
%C A319844 "even ^" means "even divisors of n in increasing order".
%C A319844 "odd v"  means "odd divisors of n in decreasing order".
%C A319844 "odd ^"  means "odd divisors of n in increasing order".
%C A319844 "t.w." means "together with".
%H A319844 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 A319844 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A319844 Triangle begins:
%e A319844    1;
%e A319844    2,  1;
%e A319844    1,  3;
%e A319844    4,  2,  1;
%e A319844    1,  5;
%e A319844    6,  2,  1,  3;
%e A319844    1,  7;
%e A319844    8,  4,  2,  1;
%e A319844    1,  3,  9;
%e A319844   10,  2,  1,  5;
%e A319844    1, 11;
%e A319844   12,  6,  4,  2,  1,  3;
%e A319844    1, 13;
%e A319844   14,  2,  1,  7;
%e A319844    1,  3,  5, 15;
%e A319844   16,  8,  4,  2,  1;
%e A319844    1, 17;
%e A319844   18,  6,  2,  1,  3,  9;
%e A319844    1, 19;
%e A319844   20, 10,  4,  2,  1,  5;
%e A319844    1,  3,  7, 21;
%e A319844   22,  2,  1, 11;
%e A319844    1, 23;
%e A319844   24, 12,  8,  6,  4,  2,  1,  3;
%e A319844    1,  5, 25;
%e A319844   26,  2,  1, 13;
%e A319844    1,  3,  9, 27;
%e A319844   28, 14,  4,  2,  1,  7;
%e A319844 ...
%e A319844 For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. The even divisors of 12 in decreasing order are [12, 6, 4, 2], and the odd divisors of 12 in increasing order are [1, 3], so the 12th row of triangle is [12, 6, 4, 2, 1, 3].
%e A319844 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 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. Then, 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. Finally all numbers c obtained are [12, 6, 4, 2, 1, 3] equaling the 12th row of triangle.
%o A319844 (PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->!(x%2), d)), select(x->(x%2), d));
%o A319844 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y A319844 Row sums give A000203.
%Y A319844 Row n has length A000005(n).
%Y A319844 Right border gives A000265.
%Y A319844 Other permutations of A027750 are A056538, A210959, A299481, A299483, A319845, A319846, A319847, A319848, A319849.
%Y A319844 Cf. A001227, A183063, A299480, A299485.
%K A319844 nonn,tabf
%O A319844 1,2
%A A319844 _Omar E. Pol_, Sep 29 2018