A129258 Rectangular array, read by descending antidiagonals: a prime separator array.

1, 2, 3, 4, 6, 5, 7, 12, 10, 8, 9, 21, 20, 16, 11, 13, 27, 35, 32, 22, 17, 15, 39, 45, 56, 44, 28, 17, 18, 45, 65, 72, 77, 56, 34, 19, 23, 54, 75, 104, 99, 98, 68, 38, 24, 25, 69, 90, 120, 143, 126, 119, 76, 48, 26, 29, 75, 115, 144, 165, 182, 153, 133, 96, 52, 30, 31, 87, 125

Offset

1,2

Comments

Every prime is in row 1 or column 1 but not both. Row 1, A129259, contains the primes 2,7,13,23,29,31. Column 1, A129260, contains the primes 3,5,11,17,19,37. For n>=2, the determinant of every n X n submatrix is zero. T(n,1)>T(1,n) for n>=2; is T(n,1)=T(1,n)+1 for infinitely many n? The least positive integer not in the array is 45.

Links

Table of n, a(n) for n=1..69.

Formula

T(1,1)=1. For n>=1, let S(n)={(i,j): 1<=i<=n and 1<=j<=n}. Once T(i,j) is defined on S(n), define T(1,n+1)=least positive integer (l.p.i.) not among T(i,j) for (i,j) in S(n); T(n+1,1)=l.p.i. not among T(i,j) for (i,j) in S(n) and not T(1,n+1); T(m,n+1)=T(m,1)*T(1,n+1) for m=2,3,...,n+1; T(n+1,m)=T(n+1,1)*T(1,m) for m=2,3,...,n+1.

Example

Corner
    1   2    4    7    9   13   15   18   23   25   29
    3   6   12   21   27   39   45   54   69   75   87
    5  10   20   35   45   65   75   90  115  125  145
    8  16   32   56   72  104  120  144  184  200  232
   11  22   44   77   99  143  165  198  253  275  319
   14  28   56   98  126  182  210  252  322  350  406
   17  34   68  119  153  221  255  306  391  425  493
   19  38   76  133  171  247  285  342  437  475  551
   24  48   96  168  216  312  360  432  552  600  696
   26  52  104  182  234  338  390  468  598  650  754
   30  60  120  210  270  390  450  540  690  750  870
   33  66  132  231  297  429  495  594  759  825  957

Mathematica

mex[list_] := First[Complement[Range[Max[list] + 1], list]];
a129258[rowL_] := Module[{T, S}, T[1, 1] = 1;
   S[n_] := Flatten[Table[T[i, j], {i, n}, {j, n}]];
   Do[T[1, n] = mex[S[n - 1]];
    T[n, 1] = mex[Append[S[n - 1], T[1, n]]];
    Do[T[m, n] = T[m, 1]*T[1, n];
     T[n, m] = T[n, 1]*T[1, m], {m, 2, n}], {n, 2, rowL}];
     Table[T[i, j], {i, rowL}, {j, rowL}]];
t = a129258[20];
Grid[t]  (* array *)
w[n_, k_] := t[[n]][[k]];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Jan 1 2025 *)

Crossrefs

Cf. A000040, A129259, A129260.

Sequence in context: A374818 A361927 A360633 * A256988 A104650 A083179

Adjacent sequences: A129255 A129256 A129257 * A129259 A129260 A129261

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 06 2007

Extensions

Edited by Clark Kimberling, Jan 11 2025

Status

approved