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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. 2
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, 32, 50, 7, 64, 56, 48, 54, 45, 66, 27, 44, 81, 72, 63, 70, 60, 84, 55, 78, 33, 100, 90, 80, 88, 77, 104, 91, 98, 65, 22, 121, 110, 99, 108, 96, 126, 112, 170, 105, 52 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the natural numbers, A000027.
Every prime (A000040) occurs in column 1.
Row 1: A000290 (squares)
Row 2: A002378) (oblong numbers)
Row 3: A005563
Row 4: A028552 (for n>=2)
LINKS
FORMULA
row 1: k^2 for k>=1
row 2: k*(k+1) for k>=1
row 3: k*k+2) for k>=3
row 4: k*(k+3) for k>=2
row 5: k*(k+4) for k >=3
row 6: k*(k+5) for k>=5
row 7: k*(k+6) for k>=7
EXAMPLE
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.
Northwest corner:
1 4 9 16 25 36 49 64 81 10
2 6 12 20 30 42 56 72 90 110
3 8 15 24 35 48 63 80 99 120
10 18 28 40 54 70 88 108 130 154
5 21 32 45 60 77 96 117 140 165
14 50 66 84 104 126 160 176 204 234
7 27 55 91 112 135 160 187 216 247
44 78 98 170 198 228 260 294 330 368
MATHEMATICA
d[n_] := Divisors[n]; k[n_] := Length[d[n]]; x[n_, i_] := d[n][[i]];
a[n_] := If[OddQ[k[n]], 0, x[n, k[n]/2 + 1] - x[n, k[n]/2]]
t = Table[a[j], {j, 1, 30000}];
r[n_] := Flatten[Position[t, n]]; v[n_, k_] := r[n][[k]];
w = Table[v[n, k], {n, 0, 10}, {k, 1, 10}];
TableForm[w] (* A285089, array *)
Table[v[n - k, k], {n, 0, 60}, {k, n, 1, -1}] // Flatten (* A285089, sequence *)
CROSSREFS
Sequence in context: A283941 A285090 A114577 * A191426 A182702 A282600
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Apr 13 2017
STATUS
approved

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)