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 A319846 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 decreasing order. 7
 1, 1, 2, 3, 1, 1, 4, 2, 5, 1, 3, 1, 6, 2, 7, 1, 1, 8, 4, 2, 9, 3, 1, 5, 1, 10, 2, 11, 1, 3, 1, 12, 6, 4, 2, 13, 1, 7, 1, 14, 2, 15, 5, 3, 1, 1, 16, 8, 4, 2, 17, 1, 9, 3, 1, 18, 6, 2, 19, 1, 5, 1, 20, 10, 4, 2, 21, 7, 3, 1, 11, 1, 22, 2, 23, 1, 3, 1, 24, 12, 8, 6, 4, 2, 25, 5, 1, 13, 1, 26, 2, 27, 9, 3, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. We can find the n-th row of the triangle as follows: Consider only the semicircumferences that contain the point [n,0]. 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. Then, 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 (see example). Sequences of the same family are shown below: ----------------------------------- Triangle Order of divisors of n ----------------------------------- A299481 odd v t.w. even ^ A299483 odd ^ t.w. even v A319844 even v t.w. odd ^ A319845 even ^ t.w. odd v This seq. odd v t.w. even v A319847 odd ^ t.w. even ^ A319848 even v t.w. odd v A319849 even ^ t.w. odd ^ ----------------------------------- In the above table we have that: "even v" means "even divisors of n in decreasing order". "even ^" means "even divisors of n in increasing order". "odd v" means "odd divisors of n in decreasing order". "odd ^" means "odd divisors of n in increasing order". "t.w." means "together with". LINKS Table of n, a(n) for n=1..95. Omar E. Pol, Figure 1: Geometric model of divisors with periodic curves (for n = 1..16), figure 2: Upper part, figure 3: Lower part upside down. Index entries for sequences related to divisors of numbers EXAMPLE Triangle begins: 1; 1, 2; 3, 1; 1, 4, 2; 5, 1; 3, 1, 6, 2; 7, 1; 1, 8, 4, 2; 9, 3, 1; 5, 1, 10, 2; 11, 1; 3, 1, 12, 6, 4, 2; 13, 1; 7, 1, 14, 2; 15, 5, 3, 1; 1, 16, 8, 4, 2; 17, 1; 9, 3, 1, 18, 6, 2; 19, 1; 5, 1, 20, 10, 4, 2; 21, 7, 3, 1; 11, 1, 22, 2; 23, 1; 3, 1, 24, 12, 8, 6, 4, 2; 25, 5, 1; 13, 1, 26, 2; 27, 9, 3, 1; 7, 1, 28, 14, 4, 2; ... 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 decreasing order are [12, 6, 4, 2], so the 12th row of triangle is [3, 1, 12, 6, 4, 2]. 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 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. Finally all numbers c obtained are [3, 1, 12, 6, 4, 2] equaling the 12th row of triangle. MATHEMATICA Table[With[{d=Divisors[n]}, Join[Reverse[Select[d, OddQ]], Reverse[Select[d, EvenQ]]]], {n, 30}]//Flatten (* Harvey P. Dale, Mar 10 2023 *) PROG (PARI) row(n) = my(d=divisors(n)); concat(Vecrev(select(x->(x%2), d)), Vecrev(select(x->!(x%2), d))); lista(nn) = {for (n=1, nn, my(r = row(n)); for (k=1, #r, print1(r[k], ", ")); ); } \\ Michel Marcus, Jan 17 2019 CROSSREFS Row sums give A000203. Row n has length A000005(n). Other permutations of A027750 are A056538, A210959, A299481, A299483, A319844, A319845, A319847, A319848, A319849. Cf. A001227, A183063, A299480, A299485. Sequence in context: A023570 A256989 A275214 * A214690 A238878 A011249 Adjacent sequences: A319843 A319844 A319845 * A319847 A319848 A319849 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Sep 29 2018 STATUS approved

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