A346459 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).

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

2

Comments

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.

If n is an element of A007916, then T(n,k) = 1 if and only if k is a perfect power of n^2.

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).

Links

Table of n, a(n) for n=2..79.

Formula

T(d^(2x), d^(2y-1)) = 1 for all positive integers d > 1, x, y.

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.

Example

Triangle T(n,k) begins:
   n\k 2   3   4   5   6   7   8   9   10   11  ...
   2   0
   3   0   0
   4   1   0   0
   5   0   0   0   0
   6   0   0   0   0   0
   7   0   0   0   0   0   0
   8   0   0   1   0   0   0   0
   9   0   1   0   0   0   0   0   0
  10   0   0   0   0   0   0   0   0   0
  11   0   0   0   0   0   0   0   0   0   0
  ...
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.

Prog

(Python)
def T(n, k):
    parity_check = [False]
    i = 0
    while True:
        while not n % k:
            n /= k
            parity_check[i] = not parity_check[i]
        if k % n:
            return 0
        elif n == 1:
            x, y = True, not parity_check[0]
            for j in range(1, i + 1):
                x, y = y, x ^ (y and parity_check[j])
            return y + 0
        else:
            n, k = k, n
            parity_check.append(False)
            i += 1
print([T(n, k) for n in range(2, 14) for k in range(2, n + 1)])
(Python)
def T(n, k):
    nk = n*k
    is_odd = 0
    while True:
        while not n % k:
            n /= k
        if k % n:
            return 0
        elif n == 1:
            while not nk % k:
                nk /= k
                is_odd = 0 if is_odd else 1
            return is_odd
        else:
            n, k = k, n
print([T(n, k) for n in range(2, 14) for k in range(2, n + 1)])

Crossrefs

Cf. A000302, A001019, A001025, A004171, A007814, A007916, A009969, A009980, A013708-A013715, A052409, A052410, A087752, A089357, A089683, A098608, A346460, A346461.

Sequence in context: A194670 A130543 A185013 * A193243 A281302 A369426

Adjacent sequences: A346456 A346457 A346458 * A346460 A346461 A346462

Keyword

nonn,tabl

Author

M. Eren Kesim, Jul 19 2021

Status

approved