A095833 -id:A095833 - cp's OEIS frontend

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.

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

Offset: 1

Clark Kimberling, Apr 13 2017

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)

			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

Clark Kimberling, Antidiagonals n = 1..60, flattened
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A000040, A000290, A095833, A163280, A285090.

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 *)

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

A357279 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.

Original entry on oeis.org

1, 2, 43, 2610, 312081, 61825050, 18318396195, 7586241152490, 4184711271725985, 2965919152834367730, 2626408950849351178875

Offset: 0

Stefano Spezia, Sep 25 2022

The n X n matrix M is the n-th principal submatrix of A002024 considered as an array, and it is singular for n > 2.

			a(2) = 43 because the hafnian of
    1  2  3  4
    2  3  4  5
    3  4  5  6
    4  5  6  7
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 43.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A002024, A002415 (absolute value of the coefficient of x^(n-2) in the characteristic polynomial of M(n)), A095833 (k-th super- and subdiagonal sums of the matrix M(n)), A204248 (permanent of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483, A356484.

M[i_, j_, n_]:=Part[Part[Table[r+c-1,{r,n},{c,n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, i+j-1);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A285090 Rectangular array by antidiagonals: the array formed by arranging the rows of A285089 so that the first column is strictly increasing.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 21, 7, 36, 30, 24, 32, 27, 10, 49, 42, 35, 45, 55, 18, 11, 64, 56, 48, 60, 91, 28, 39, 13, 81, 72, 63, 77, 112, 40, 75, 85, 14, 100, 90, 80, 96, 135, 54, 119, 133, 50, 17, 121, 110, 99, 117, 160, 70, 171, 189, 66

Offset: 1

Clark Kimberling, Apr 13 2017

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. For each row, there is a nonnegative integer h such that all but finitely many initial entries are of the form k*(k+h).

			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
5   21  32   45   60   77   96   117  140  165
7   27  55   91   112  135  160  187  216  247
10  18  28   40   54   70   88   108  130  154
11  39  75   119  171  200  231  264  299  375
13  85  133  189  253  325  364  405  448  493

Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A000040, A000290, A095833, A163280, A285089.

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, 20}, {k, 1, 20}];
y = SortBy[w, First]; v[n_, k_] := y[[n, k]];
w = TableForm[Table[v[n, k], {n, 1, 10}, {k, 1, 10}]]
Table[v[n + 1 - k, k], {n, 1, 15}, {k, n, 1, -1}] // Flatten

A357419 a(n) is the hafnian of the 2n X 2n symmetric Pascal matrix defined by M[i, j] = A007318(i + j - 2, i - 1).

Original entry on oeis.org

1, 1, 17, 4929, 23872137, 1901611778409, 2469317979267366913, 52019468048773355156225921, 17726418489020770628047341494927089, 97518325438289444681986165275143492027985129, 8648473129650550498122567373327602114148485950241817345

Offset: 0

Stefano Spezia, Sep 27 2022

			a(2) = 17 because the hafnian of
    1,  1,  1,   1
    1,  2,  3,   4
    1,  3,  6,  10
    1,  4, 10,  20
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 17.

Eric Weisstein's World of Mathematics, Pascal Matrix
Wikipedia, Hafnian
Wikipedia, Pascal matrix
Wikipedia, Symmetric matrix

Cf. A007318.
Cf. A006134 (trace of M(n)), A095833 (k-th super- and subdiagonal sums of M(n)), A320845 (permanent of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483, A356484.

M[i_, j_, n_]:=Part[Part[Table[Binomial[r+c-2,r-1], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

a(6)-a(10) from Pontus von Brömssen, Oct 14 2023

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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.

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A357279 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i, j] = i + j - 1.

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A285090 Rectangular array by antidiagonals: the array formed by arranging the rows of A285089 so that the first column is strictly increasing.

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A357419 a(n) is the hafnian of the 2n X 2n symmetric Pascal matrix defined by M[i, j] = A007318(i + j - 2, i - 1).

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