cp's OEIS Frontend

A319849 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 increasing order.

%I A319849 #29 Jan 18 2019 04:43:47
%S A319849 1,2,1,1,3,2,4,1,1,5,2,6,1,3,1,7,2,4,8,1,1,3,9,2,10,1,5,1,11,2,4,6,12,
%T A319849 1,3,1,13,2,14,1,7,1,3,5,15,2,4,8,16,1,1,17,2,6,18,1,3,9,1,19,2,4,10,
%U A319849 20,1,5,1,3,7,21,2,22,1,11,1,23,2,4,6,8,12,24,1,3,1,5,25,2,26,1,13,1,3,9,27
%N A319849 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 increasing order.
%C A319849 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 A319849 We can find the n-th row of the triangle as follows:
%C A319849 Consider only the semicircumferences that contain the point [n,0].
%C A319849 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 A319849 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 A319849 Sequences of the same family are shown below:
%C A319849 -----------------------------------
%C A319849 Triangle    Order of divisors of n
%C A319849 -----------------------------------
%C A319849 A299481      odd  v   t.w.  even ^
%C A319849 A299483      odd  ^   t.w.  even v
%C A319849 A319844      even v   t.w.  odd  ^
%C A319849 A319845      even ^   t.w.  odd  v
%C A319849 A319846      odd  v   t.w.  even v
%C A319849 A319847      odd  ^   t.w.  even ^
%C A319849 A319848      even v   t.w.  odd  v
%C A319849 This seq.    even ^   t.w.  odd  ^
%C A319849 -----------------------------------
%C A319849 In the above table we have that:
%C A319849 "even v" means "even divisors of n in decreasing order".
%C A319849 "even ^" means "even divisors of n in increasing order".
%C A319849 "odd v"  means "odd divisors of n in decreasing order".
%C A319849 "odd ^"  means "odd divisors of n in increasing order".
%C A319849 "t.w." means "together with".
%H A319849 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 A319849 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A319849 Triangle begins:
%e A319849    1;
%e A319849    2,  1;
%e A319849    1,  3;
%e A319849    2,  4,  1;
%e A319849    1,  5;
%e A319849    2,  6,  1,  3;
%e A319849    1,  7;
%e A319849    2,  4,  8,  1;
%e A319849    1,  3,  9;
%e A319849    2, 10,  1,  5;
%e A319849    1, 11;
%e A319849    2,  4,  6, 12,  1,  3;
%e A319849    1, 13;
%e A319849    2, 14,  1,  7;
%e A319849    1,  3,  5, 15;
%e A319849    2,  4,  8, 16,  1;
%e A319849    1, 17;
%e A319849    2,  6, 18,  1,  3,  9;
%e A319849    1, 19;
%e A319849    2,  4, 10, 20,  1,  5;
%e A319849    1,  3,  7, 21;
%e A319849    2, 22,  1, 11;
%e A319849    1, 23;
%e A319849    2,  4,  6,  8, 12, 24,  1,  3;
%e A319849    1,  5, 25;
%e A319849    2, 26,  1, 13;
%e A319849    1,  3,  9, 27;
%e A319849    2,  4, 14, 28,  1,  7;
%e A319849 ...
%e A319849 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 increasing order are [1, 3], so the 12th row of triangle is [2, 4, 6, 12, 1, 3].
%e A319849 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 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 [2, 4, 6, 12, 1, 3] equaling the 12th row of triangle.
%o A319849 (PARI) row(n) = my(d=divisors(n)); concat(select(x->!(x%2), d), select(x->(x%2), d));
%o A319849 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%Y A319849 Row sums give A000203.
%Y A319849 Row n has length A000005(n).
%Y A319849 Column 1 gives A000034.
%Y A319849 Right border gives A000265.
%Y A319849 Other permutations of A027750 are A056538, A210959, A299481, A299483, A319844, A319845, A319846, A319847, A319848.
%Y A319849 Cf. A001227, A183063, A299480, A299485.
%K A319849 nonn,tabf
%O A319849 1,2
%A A319849 _Omar E. Pol_, Sep 29 2018