OFFSET
1,2
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, ...).
LINKS
Rémy Sigrist, PARI program for A284145
Rémy Sigrist, Representation of prime numbers among the first 100 rows
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Bob Selcoe, Mar 20 2017
STATUS
approved