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%I #19 Oct 07 2013 13:27:28
%S 1,2,4,7,11,13,15,18,19,20,25,26,28,35,39,40,44,45,47,48,49,53,54,55,
%T 56,60,61,63,67,71,72,74,76,79,81,83,87,100,104,106,107,109,112,116,
%U 118,126,127,128,135,139,143
%N Numbers that are the distance between a square and the next larger cube.
%C This is the range of the sequence A181138 (= least k>0 such that n^2+k is a cube). Note that this is not the same as A087285 = range of A077116 = difference between a cube and the next smaller square: If n^2+k=y^3 is the smallest cube above n^2, then n^2 is not necessarily the largest square below y^3, e.g., 9+18=27=3^3 is the least cube above 9=3^2, but 25=5^2 is the largest square below 27. Therefore the number 18 is in this sequence, but not in A087285.
%C See A077116 and A181138 and A179386 for motivations.
%C Apart from the leading 1, this is a subsequence of A106265, which does not require the square to be the next smaller one: For example, 23 = 27 - 4 = 3^3 - 2^2 is in A106265 but not in this sequence. A165288 is a subsequence of this one, except for the initial term.
%e a(1) = 1 = 1^3-0^2 (but this is the only solution to y^3-x^2=1).
%e a(2) = 2 = 27-25 (= 3^3-5^2), and this is the only solution to y^3-x^2=2.
%e The number 3 is not in the sequence since there are no x,y > 0 such that y^3-x^2=3.
%e a(3) = 4 = 8-4 (= 2^3-2^2) = 125-121 (= 5^3-11^2); these are the only two solutions to y^3-x^2=4, for all x>11, the minimal positive y^3-x^2 is 7.
%Y Cf. A087285, A087286, A088017, A081121, A081120, A077116, A065733.
%K nonn,more
%O 1,2
%A _M. F. Hasler_, Sep 26 2013
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