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.

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

Author

Bob Selcoe, Mar 22 2017

Keywords

• nonn
• tabl

Comments

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, ...).

Examples

			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.
		

Crossrefs

Cf. A284145.