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A229618
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Numbers that are the distance between a square and the next larger cube.
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6
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1, 2, 4, 7, 11, 13, 15, 18, 19, 20, 25, 26, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 67, 71, 72, 74, 76, 79, 81, 83, 87, 100, 104, 106, 107, 109, 112, 116, 118, 126, 127, 128, 135, 139, 143
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OFFSET
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1,2
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COMMENTS
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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.
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.
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LINKS
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EXAMPLE
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a(1) = 1 = 1^3-0^2 (but this is the only solution to y^3-x^2 = 1).
a(2) = 2 = 27-25 (= 3^3-5^2), and this is the only solution to y^3-x^2 = 2.
The number 3 is not in the sequence since there are no x, y > 0 such that y^3-x^2 = 3.
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.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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