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A087285
Possible differences between a cube and the next smaller square.
12
2, 4, 7, 11, 13, 15, 19, 20, 26, 28, 35, 39, 40, 45, 47, 48, 49, 53, 55, 56, 60, 63, 67, 74, 76, 79, 81, 83, 100, 104, 107, 109, 116, 127, 135, 139, 146, 147, 148, 150, 152, 155, 170, 174, 180, 184, 186, 191, 193, 200, 207, 212, 215, 216, 233, 235, 242, 244, 249
OFFSET
1,1
COMMENTS
Sequence and program were provided by Ralf Stephan Aug 28 2003.
Comment from David W. Wilson, Jan 05 2009: I believe there is an algorithm for solving x^3 - y^2 = k, which should have a finite number of solutions for any k. That means that we should in principle be able to compute this sequence.
Up to the initial 0 in A165288, these two sequences appear to be the same, but according to its current definition, A165288 should be the same as the (different) sequence A229618 = the range of the sequence A181138 (= least k>0 such that n^2+k is a cube): 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., 18 is in A181138 and A229618, since 9+18=27 is the least cube above 9=3^2, but 25=5^2 is the largest square below 27. - M. F. Hasler, Oct 05 2013
REFERENCES
See under A081121.
EXAMPLE
a(1)=2 because the next smaller square below 3^3=27 is 5^2=25.
PROG
(PARI) v=vector(200):for(n=2, 10^7, t=n^3:s=sqrtint(t)^2: if(s==t, s=sqrtint(t-1)^2):tt=t-s: if(tt>0&&tt<=200&&!v[tt], v[tt]=n)):for(k=1, 200, if(v[k], print1(k", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Sep 18 2003
STATUS
approved