A229618 Numbers that are the distance between a square and the next larger cube.

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

Offset

1,2

Comments

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.

See A077116 and A181138 and A179386 for motivations.

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.

Links

Table of n, a(n) for n=1..51.

Example

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.

Crossrefs

Cf. A087285, A087286, A088017, A081121, A081120, A077116, A065733.

Sequence in context: A127575 A240106 A206853 * A087285 A107791 A181518

Adjacent sequences: A229615 A229616 A229617 * A229619 A229620 A229621

Keyword

nonn,more

Author

M. F. Hasler, Sep 26 2013

Status

approved