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%I #29 Jan 26 2022 08:22:01
%S 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0
%N Triangle read by rows: T(n,k) = 0 if all positive integers can be colored with two colors without any positive integer x being the same color as n*x or k*x; otherwise, T(n,k) = 1 (for 2 <= k <= n).
%C T(n,k) = 1 if and only if there exists at least one pair of positive integers (x, y) such that n^x = k^y and x+y is odd. Otherwise, T(n,k) = 0.
%C If n is an element of A007916, then T(n,k) = 1 if and only if k is a perfect power of n^2.
%C T(n,k) = 1 if and only if there exists a positive integer x for which A052410(n)^x = k and A007814(A052409(n)) != A007814(x).
%F T(d^(2x), d^(2y-1)) = 1 for all positive integers d > 1, x, y.
%F T(A000302(n), A004171(k)) = T(A001019(n), A013708(k)) = T(A001025(n), A013709(k)) = T(A009969(n), A013710(k)) = T(A009980(n), A013711(k)) = T(A087752(n), A013712(k)) = T(A089357(n), A013713(k)) = T(A089683(n), A013714(k)) = T(A098608(n), A013715(k)) = 1 for all n >= 1, k >= 0.
%e Triangle T(n,k) begins:
%e n\k 2 3 4 5 6 7 8 9 10 11 ...
%e 2 0
%e 3 0 0
%e 4 1 0 0
%e 5 0 0 0 0
%e 6 0 0 0 0 0
%e 7 0 0 0 0 0 0
%e 8 0 0 1 0 0 0 0
%e 9 0 1 0 0 0 0 0 0
%e 10 0 0 0 0 0 0 0 0 0
%e 11 0 0 0 0 0 0 0 0 0 0
%e ...
%e If we color all positive integers whose 2-adic order and 3-adic order add up to an even number in color A and the rest in color B, every positive integer will be a different color from its double and triple. Therefore, T(3, 2) = 0.
%o (Python)
%o def T(n, k):
%o parity_check = [False]
%o i = 0
%o while True:
%o while not n % k:
%o n /= k
%o parity_check[i] = not parity_check[i]
%o if k % n:
%o return 0
%o elif n == 1:
%o x, y = True, not parity_check[0]
%o for j in range(1, i + 1):
%o x, y = y, x ^ (y and parity_check[j])
%o return y + 0
%o else:
%o n, k = k, n
%o parity_check.append(False)
%o i += 1
%o print([T(n, k) for n in range(2, 14) for k in range(2, n + 1)])
%o (Python)
%o def T(n, k):
%o nk = n*k
%o is_odd = 0
%o while True:
%o while not n % k:
%o n /= k
%o if k % n:
%o return 0
%o elif n == 1:
%o while not nk % k:
%o nk /= k
%o is_odd = 0 if is_odd else 1
%o return is_odd
%o else:
%o n, k = k, n
%o print([T(n, k) for n in range(2, 14) for k in range(2, n + 1)])
%Y Cf. A000302, A001019, A001025, A004171, A007814, A007916, A009969, A009980, A013708-A013715, A052409, A052410, A087752, A089357, A089683, A098608, A346460, A346461.
%K nonn,tabl
%O 2
%A _M. Eren Kesim_, Jul 19 2021
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