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A319847
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Irregular triangle read by rows in which row n lists the odd divisors of n in increasing order together with the even divisors of n in increasing order.
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7
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1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 3, 2, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 5, 2, 10, 1, 11, 1, 3, 2, 4, 6, 12, 1, 13, 1, 7, 2, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 3, 9, 2, 6, 18, 1, 19, 1, 5, 2, 4, 10, 20, 1, 3, 7, 21, 1, 11, 2, 22, 1, 23, 1, 3, 2, 4, 6, 8, 12, 24, 1, 5, 25, 1, 13, 2, 26, 1, 3, 9, 27
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OFFSET
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1,3
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COMMENTS
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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 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.
Then, 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 (see example).
Sequences of the same family are shown below:
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Triangle Order of divisors of n
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This seq. odd ^ t.w. even ^
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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".
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LINKS
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EXAMPLE
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Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 5;
1, 3, 2, 6;
1, 7;
1, 2, 4, 8;
1, 3, 9;
1, 5, 2, 10;
1, 11;
1, 3, 2, 4, 6, 12;
1, 13;
1, 7, 2, 14;
1, 3, 5, 15;
1, 2, 4, 8, 16;
1, 17;
1, 3, 9, 2, 6, 18;
1, 19;
1, 5, 2, 4, 10, 20;
1, 3, 7, 21;
1, 11, 2, 22;
1, 23;
1, 3, 2, 4, 6, 8, 12, 24;
1, 5, 25;
1, 13, 2, 26;
1, 3, 9, 27;
1, 7, 2, 4, 14, 28;
...
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12]. The odd divisors of 12 in increasing order are [1, 3], and the even divisors of 12 in increasing order are [2, 4, 6, 12], so the 12th row of triangle is [1, 3, 2, 4, 6, 12].
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 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. Then, 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. Finally all numbers c obtained are [1, 3, 2, 4, 6, 12] equaling the 12th row of triangle.
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PROG
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(PARI) row(n) = my(d=divisors(n)); concat(select(x->(x%2), d), 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
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CROSSREFS
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A permutation of A027750 from which first differs at a(12) = T(6,2).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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