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A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as n*x or k*x (for n >= 2).

%I A346460 #24 Dec 19 2024 11:48:12
%S A346460 4,16,9,64,81,2,256,729,8,25,1024,6561,16,625,36,4096,59049,32,15625,
%T A346460 1296,49,16384,531441,128,390625,46656,2401,4,65536,4782969,256,
%U A346460 9765625,1679616,117649,16,3,262144,43046721,512,244140625,60466176,5764801,64,27,100
%N A346460 Square array read by downward antidiagonals in which row n lists all numbers k for which all positive integers cannot be colored with two colors without any positive integer x being the same color as n*x or k*x (for n >= 2).
%C A346460 Row n lists all positive integers k for which there exists at least one pair of positive integers (x, y) such that n^x = k^y and x+y is odd.
%C A346460 If n is an element of A007916, then row n lists all perfect powers of n^2.
%C A346460 A positive integer k is in row n if and only if there exists a positive integer x for which A052410(n)^x = k and A007814(A052409(n)) != A007814(x).
%H A346460 M. Eren Kesim, <a href="/A346460/a346460_2.txt">First Python program for A346460</a>
%H A346460 M. Eren Kesim, <a href="/A346460/a346460_3.txt">Second Python program for A346460</a>
%e A346460 Table begins:
%e A346460      4,    16,      64,       256,        1024,          4096,           16384, ...
%e A346460      9,    81,     729,      6561,       59049,        531441,         4782969, ...
%e A346460      2,     8,      16,        32,         128,           256,             512, ...
%e A346460     25,   625,   15625,    390625,     9765625,     244140625,      6103515625, ...
%e A346460     36,  1296,   46656,   1679616,    60466176,    2176782336,      7836416409, ...
%e A346460     49,  2401,  117649,   5764801,   282475249,   13841287201,    678223072849, ...
%e A346460      4,    16,      64,       256,        1024,          4096,           16384, ...
%e A346460      3,    27,      81,       243,        2187,          6561,           19683, ...
%e A346460    100, 10000, 1000000, 100000000, 10000000000, 1000000000000, 100000000000000, ...
%o A346460 (Python) # See links.
%Y A346460 First few rows are A000302, A001019, A346461, A009969, A009980, and A087752.
%Y A346460 Cf. A007814, A007916, A013708-A013715, A052409, A052410, A098608, A346459.
%K A346460 nonn,tabl
%O A346460 2,1
%A A346460 _M. Eren Kesim_, Aug 25 2021