A284145 -id:A284145 - cp's OEIS frontend

A366030 Square array A(n, k), n, k > 0, read and filled by upwards antidiagonals the greedy way with distinct positive integers such that any two distinct terms in the same row or column or antidiagonal are coprime.

1, 2, 3, 5, 7, 4, 9, 8, 11, 13, 17, 19, 21, 23, 25, 29, 31, 37, 41, 27, 43, 47, 53, 59, 10, 61, 67, 49, 71, 55, 73, 57, 79, 83, 89, 97, 77, 101, 103, 107, 16, 109, 113, 65, 127, 131, 137, 85, 139, 91, 121, 149, 151, 157, 163, 167, 169, 173, 179, 181, 6, 115, 119, 191, 193, 197

Offset: 1

Rémy Sigrist, Sep 26 2023

This sequence is a variant of A284145 (with one less constraint).
All the prime numbers appear in the sequence, in ascending order.
For any prime number p, the first multiple of p in the sequence is p.
Will every positive integer appear in the sequence?

			Array A(n, k) begins:
  n\k|   1    2    3    4    5    6    7    8    9   10
  ---+-------------------------------------------------
    1|   1    3    4   13   25   43   49   97  127  163
    2|   2    7   11   23   27   67   89   65  157  193
    3|   5    8   21   41   61   83  113  151  191  221
    4|   9   19   37   10   79  109  149  119  239  281
    5|  17   31   59   57   16  121  115  233  203  347
    6|  29   53   73  107   91    6  229  277  337  125
    7|  47   55  103  139  181  161   12  331  323  463
    8|  71  101   85  179  209  271  317   18  403  259
    9|  77  137  173  227  269   95  377  461   24  613
   10| 131  169  223  187  313  397  457  437  185   32

Rémy Sigrist, Colored representation of the array for n, k <= 500 (grayish pixels correspond to prime numbers)
Rémy Sigrist, PARI program

Cf. A284145, A366031, A366303.

PARI
```
See Links section.
```

A284189 Square array T(n,k) read by upward antidiagonals: each term is the least positive integer not yet appearing in the array that is coprime to all the terms in its associated row, column, diagonal and antidiagonal.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 9, 11, 13, 17, 19, 8, 23, 15, 29, 31, 25, 37, 14, 41, 43, 47, 53, 21, 59, 61, 67, 55, 49, 71, 73, 79, 83, 27, 89, 97, 101, 103, 85, 107, 16, 77, 109, 113, 127, 121, 131, 137, 139, 35, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 95, 199, 187, 161

Offset: 1

Bob Selcoe, Mar 22 2017

A variant of triangle A284145: this array is built by antidiagonals originating at T(n,1), while A284145 is built by rows.
Conjecture 1: The array is a permutation of the natural numbers.
Conjecture 2: The prime factors of all the terms in each individual row, column and diagonal are permutations of the prime numbers (except the middle diagonal and the first row and column, which obviously also contain 1).
Let S be a set of terms whose members have certain specified characteristics (e.g., even numbers or prime numbers). Sets S whose members appear in due course in ascending order include:
(a) Prime numbers (so 2 appears first, followed by 3, 5, 7, 11, ...);
(b) Numbers which have exactly the same prime factors (so for example: {6, 12, 18, 24, 36, 48, 54, 72, ...} appear ascending order because their prime factors are {2,3});
(c) Powers of prime(j), because they are a subcategory of (b) (so for example: 5 appears first, followed by 25, 125, 625, 3125, ...).

			Array begins:
    1,   3,   4,  17,  29,  43,  55,  97, 127, 167, ...
    2,   7,  13,  15,  41,  67,  89, 113, 163, 187, ...
    5,  11,  23,  14,  61,  27, 109, 157, 199, 211, ...
    9,   8,  37,  59,  83,  77, 151,  95, 221, 223, ...
   19,  25,  21,  79,  16, 149, 197, 227, 229, 233, ...
   31,  53,  73, 107,  35, 193, 239, 241,  22,  39, ...
   47,  71,  85, 139, 191, 251,  57, 257, 263, 203, ...
   49, 103, 137, 181, 209, 269, 271, 277, 115, 281, ...
  101, 131, 179, 283, 293, 289, 307,  81, 311, 313, ...
  121, 173, 317, 299, 111,  32, 331, 337, 347, 125, ...
T(6,5) = 35 because a term with prime factor 2 already appears in the diagonal (and column) to T(6,5); no terms with prime factors 5 or 7 appear in any row, column, diagonal or antidiagonal to T(6,5); and terms 5, 7, and 25 already appear in the array. Note that while no term with prime factor 3 appears in any row, column, diagonal or antidiagonal to T(6,5), no multiple of 3 < 35 can be placed there because 3, 9, 15, 21 and 27 have already appeared in the array and 11 is in its diagonal.

Cf. A284145.

cp's OEIS Frontend

A366030 Square array A(n, k), n, k > 0, read and filled by upwards antidiagonals the greedy way with distinct positive integers such that any two distinct terms in the same row or column or antidiagonal are coprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI