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A285089 Rectangular array by antidiagonals: row n is the ordered sequence of numbers k that minimize |d(n+1-k) - d(k)|, where d(i) are the divisors of n.

%I A285089 #10 Nov 22 2024 00:34:58
%S A285089 1,4,2,9,6,3,16,12,8,10,25,20,15,18,5,36,30,24,28,21,14,49,42,35,40,
%T A285089 32,50,7,64,56,48,54,45,66,27,44,81,72,63,70,60,84,55,78,33,100,90,80,
%U A285089 88,77,104,91,98,65,22,121,110,99,108,96,126,112,170,105,52
%N A285089 Rectangular array by antidiagonals: row n is the ordered sequence of numbers k that minimize |d(n+1-k) - d(k)|, where d(i) are the divisors of n.
%C A285089 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the natural numbers, A000027.
%C A285089 Every prime (A000040) occurs in column 1.
%C A285089 Row 1: A000290 (squares)
%C A285089 Row 2: A002378 (oblong numbers)
%C A285089 Row 3: A005563
%C A285089 Row 4: A028552 (for n>=2)
%H A285089 Clark Kimberling, <a href="/A285089/b285089.txt">Antidiagonals n = 1..60, flattened </a>
%H A285089 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A285089 row 1: k^2 for k>=1
%F A285089 row 2: k*(k+1) for k>=1
%F A285089 row 3: k*(k+2) for k>=3
%F A285089 row 4: k*(k+3) for k>=2
%F A285089 row 5: k*(k+4) for k>=3
%F A285089 row 6: k*(k+5) for k>=5
%F A285089 row 7: k*(k+6) for k>=7
%e A285089 Taking n = 12, the divisors are 1,2,3,4,6,12, so that for k=1..6, the numbers d(n+1-k) - d(k) are 12-1, 6-2, 4-3, 3-4, 2-6, 1-12.  Thus, the number k that minimizes |d(n+1-k) - d(k)| is 1, so that 12 appears in row 1 (with the top row as row 0), consisting of numbers for which the minimal value is 1.
%e A285089 Northwest corner:
%e A285089   1   4   9   16   25   36   49   64   81   10
%e A285089   2   6   12  20   30   42   56   72   90   110
%e A285089   3   8   15  24   35   48   63   80   99   120
%e A285089   10  18  28  40   54   70   88   108  130  154
%e A285089   5   21  32  45   60   77   96   117  140  165
%e A285089   14  50  66  84   104  126  160  176  204  234
%e A285089   7   27  55  91   112  135  160  187  216  247
%e A285089   44  78  98  170  198  228  260  294  330  368
%t A285089 d[n_] := Divisors[n]; k[n_] := Length[d[n]]; x[n_, i_] := d[n][[i]];
%t A285089 a[n_] := If[OddQ[k[n]], 0, x[n, k[n]/2 + 1] - x[n, k[n]/2]]
%t A285089 t = Table[a[j], {j, 1, 30000}];
%t A285089 r[n_] := Flatten[Position[t, n]]; v[n_, k_] := r[n][[k]];
%t A285089 w = Table[v[n, k], {n, 0, 10}, {k, 1, 10}];
%t A285089 TableForm[w] (* A285089, array *)
%t A285089 Table[v[n - k, k], {n, 0, 60}, {k, n, 1, -1}] // Flatten (* A285089, sequence *)
%Y A285089 Cf. A000027, A000040, A000290, A095833, A163280, A285090.
%K A285089 nonn,tabl,easy
%O A285089 1,2
%A A285089 _Clark Kimberling_, Apr 13 2017