A284145 Triangle read by rows T(n,k) in which each term is the least positive integer not yet appearing in the triangle 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, 21, 23, 25, 29, 31, 37, 16, 27, 41, 43, 47, 53, 35, 59, 61, 67, 49, 71, 73, 33, 79, 83, 85, 89, 97, 101, 95, 103, 14, 107, 109, 113, 121, 127, 131, 137, 139, 143, 115, 149, 151, 157, 133, 163, 65, 167, 173, 179, 6, 181, 187, 191, 193, 197

Offset: 1

Author

Bob Selcoe, Mar 20 2017

Keywords

• nonn
• tabl

Comments

Conjecture 1: The triangle is a permutation of the natural numbers.

Let F(k) and G(n) be the set of prime factors of all terms in a given column k or diagonal n (diagonal n originates at T(n,1)).

Conjecture 2: Each F(k) and G(n) is a permutation of the prime numbers (except F(1) and G(1), 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

			Triangle begins:
    1
    2   3
    5   7   4
    9  11  13  17
   19   8  21  23  25
   29  31  37  16  27  41
   43  47  53  35  59  61  67
   49  71  73  33  79  83  85  89
   97 101  95 103  14 107 109 113 121
  127 131 137 139 143 115 149 151 157 133
  163  65 167 173 179   6 181 187 191 193 197
T(7,4) = 35 because terms with prime factor 2 already appear in the diagonal (and column), and terms with prime factor 3 already appear in the diagonal (and antidiagonal) to T(7,4); no terms with prime factors 5 or 7 appear in any row, column, diagonal or antidiagonal to T(7,4); and terms 5, 7, and 25 already appear in the triangle.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..20100; rows 1..50 in flattened form
• Rémy Sigrist, PARI program for A284145
• Rémy Sigrist, Representation of prime numbers among the first 100 rows

Crossrefs

Cf. A274651.