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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.

%I A284145 #32 Mar 09 2025 06:28:21
%S A284145 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,
%T A284145 59,61,67,49,71,73,33,79,83,85,89,97,101,95,103,14,107,109,113,121,
%U A284145 127,131,137,139,143,115,149,151,157,133,163,65,167,173,179,6,181,187,191,193,197
%N 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.
%C A284145 Conjecture 1: The triangle is a permutation of the natural numbers.
%C A284145 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)).
%C A284145 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).
%C A284145 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:
%C A284145 (a) Prime numbers (so 2 appears first, followed by 3, 5, 7, 11, ...);
%C A284145 (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 A284145 (c) Powers of prime(j), because they are a subcategory of (b) (so for example: 5 appears first, followed by 25, 125, 625, 3125, ...).
%H A284145 Rémy Sigrist, <a href="/A284145/b284145.txt">Table of n, a(n) for n = 1..20100; rows 1..50 in flattened form</a>
%H A284145 Rémy Sigrist, <a href="/A284145/a284145.gp.txt">PARI program for A284145</a>
%H A284145 Rémy Sigrist, <a href="/A284145/a284145.png">Representation of prime numbers among the first 100 rows</a>
%e A284145 Triangle begins:
%e A284145     1
%e A284145     2   3
%e A284145     5   7   4
%e A284145     9  11  13  17
%e A284145    19   8  21  23  25
%e A284145    29  31  37  16  27  41
%e A284145    43  47  53  35  59  61  67
%e A284145    49  71  73  33  79  83  85  89
%e A284145    97 101  95 103  14 107 109 113 121
%e A284145   127 131 137 139 143 115 149 151 157 133
%e A284145   163  65 167 173 179   6 181 187 191 193 197
%e A284145 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.
%Y A284145 Cf. A274651.
%K A284145 nonn,tabl
%O A284145 1,2
%A A284145 _Bob Selcoe_, Mar 20 2017