This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319848 #36 Jan 18 2019 04:43:37
%S A319848 1,2,1,3,1,4,2,1,5,1,6,2,3,1,7,1,8,4,2,1,9,3,1,10,2,5,1,11,1,12,6,4,2,
%T A319848 3,1,13,1,14,2,7,1,15,5,3,1,16,8,4,2,1,17,1,18,6,2,9,3,1,19,1,20,10,4,
%U A319848 2,5,1,21,7,3,1,22,2,11,1,23,1,24,12,8,6,4,2,3,1,25,5,1,26,2,13,1,27,9,3,1
%N A319848 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 decreasing order.
%C A319848 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 A319848 We can find the n-th row of the triangle as follows:
%C A319848 Consider only the semicircumferences that contain the point [n,0].
%C A319848 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 A319848 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 A319848 Sequences of the same family are shown below:
%C A319848 -----------------------------------
%C A319848 Triangle Order of divisors of n
%C A319848 -----------------------------------
%C A319848 A299481 odd v t.w. even ^
%C A319848 A299483 odd ^ t.w. even v
%C A319848 A319844 even v t.w. odd ^
%C A319848 A319845 even ^ t.w. odd v
%C A319848 A319846 odd v t.w. even v
%C A319848 A319847 odd ^ t.w. even ^
%C A319848 This seq. even v t.w. odd v
%C A319848 A319849 even ^ t.w. odd ^
%C A319848 -----------------------------------
%C A319848 In the above table we have that:
%C A319848 "even v" means "even divisors of n in decreasing order".
%C A319848 "even ^" means "even divisors of n in increasing order".
%C A319848 "odd v" means "odd divisors of n in decreasing order".
%C A319848 "odd ^" means "odd divisors of n in increasing order".
%C A319848 "t.w." means "together with".
%H A319848 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 A319848 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 A319848 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A319848 Triangle begins:
%e A319848 1;
%e A319848 2, 1;
%e A319848 3, 1;
%e A319848 4, 2, 1;
%e A319848 5, 1;
%e A319848 6, 2, 3, 1;
%e A319848 7, 1;
%e A319848 8, 4, 2, 1;
%e A319848 9, 3, 1;
%e A319848 10, 2, 5, 1;
%e A319848 11, 1;
%e A319848 12, 6, 4, 2, 3, 1;
%e A319848 13, 1;
%e A319848 14, 2, 7, 1;
%e A319848 15, 5, 3, 1;
%e A319848 16, 8, 4, 2, 1;
%e A319848 17, 1;
%e A319848 18, 6, 2, 9, 3, 1;
%e A319848 19, 1;
%e A319848 20, 10, 4, 2, 5, 1;
%e A319848 21, 7, 3, 1;
%e A319848 22, 2, 11, 1;
%e A319848 23, 1;
%e A319848 24, 12, 8, 6, 4, 2, 3, 1;
%e A319848 25, 5, 1;
%e A319848 26, 2, 13, 1;
%e A319848 27, 9, 3, 1;
%e A319848 28, 14, 4, 2, 7, 1;
%e A319848 ...
%e A319848 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 decreasing order are [3, 1], so the 12th row of triangle is [12, 6, 4, 2, 3, 1].
%e A319848 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 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 [12, 6, 4, 2, 3, 1] equaling the 12th row of triangle.
%e A319848 From _David A. Corneth_, Jan 17 2019: (Start)
%e A319848 For n = 100, we have n even so first even divisors in decreasing order and then odd divisors in decreasing order. The even divisors of 100 are twice the divisors of 100/2 = 50. Those divisors, in decreasing order, are: 100, 50, 20, 10, 4, 2.
%e A319848 The odd divisors of 100 in decreasing order are the odd divisors of the odd part of 100, which is 25. Those divisors are 25, 5, 1.
%e A319848 Concatenating all these divisors gives: 100, 50, 20, 10, 4, 2, 25, 5, 1. (End)
%o A319848 (PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->!(x%2), d)), Vecrev(select(x->(x%2), d)));
%o A319848 lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")););} \\ _Michel Marcus_, Jan 17 2019
%o A319848 (PARI) row(n) = if(n%2==0, concat(Vecrev(divisors(n >> 1)) << 1, Vecrev(divisors(n >> valuation(n,2)))), Vecrev(divisors(n))) \\ _David A. Corneth_, Jan 17 2019
%Y A319848 Row sums give A000203.
%Y A319848 Row n has length A000005(n).
%Y A319848 Column 1 gives A000027.
%Y A319848 Right border gives A000012.
%Y A319848 Other permutations of A027750 are A056538, A210959, A299481, A299483, A319844, A319845, A319846, A319847, A319849.
%Y A319848 Cf. A001227, A056538, A183063, A210959, A299480, A299485.
%K A319848 nonn,tabf
%O A319848 1,2
%A A319848 _Omar E. Pol_, Sep 29 2018