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Infinite sequence of nonzero integers build the greedy way such that the products Product_{i = k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.
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%I #7 Mar 07 2023 07:41:45

%S -1,2,3,-3,4,-4,5,-5,6,-6,7,-7,8,-8,9,10,-10,11,-11,12,-12,13,-13,14,

%T -14,15,-15,16,17,-17,-18,19,-19,20,-20,21,-21,22,-22,23,-23,24,-24,

%U 25,26,-26,27,-27,28,-28,29,-29,30,-30,31,-31,32,-32,33,-33,34

%N Infinite sequence of nonzero integers build the greedy way such that the products Product_{i = k*2^e..(k+1)*2^e} a(i) with k, e >= 0 are all distinct; each term is minimal in absolute value and in case of a tie, preference is given to the positive value.

%C This sequence is a variant of A360305 where we allow negative values.

%C In order for the sequence to be infinite, the value 1 is forbidden.

%C We can arrange the terms of the sequence as the leaves of a perfect infinite binary tree, the products with e > 0 corresponding to parent nodes; each node will contain a different value and all values except 0 and 1 will appear in the tree.

%H Rémy Sigrist, <a href="/A361234/a361234.gp.txt">PARI program</a>

%e The first terms (at the bottom of the tree) alongside the corresponding products are:

%e -73156608000

%e ---------------------------------

%e 7200 -10160640

%e ----------------- -----------------

%e 18 400 1764 -5760

%e --------- --------- --------- ---------

%e -2 -9 -16 -25 -36 -49 -64 90

%e ----- ----- ----- ----- ----- ----- ----- -----

%e -1 2 3 -3 4 -4 5 -5 6 -6 7 -7 8 -8 9 10

%o (PARI) See Links section.

%Y Cf. A360305, A361144.

%K sign

%O 1,2

%A _Rémy Sigrist_, Mar 05 2023