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A045980 Numbers of the form x^3 + y^3 or x^3 - y^3. 12
0, 1, 2, 7, 8, 9, 16, 19, 26, 27, 28, 35, 37, 54, 56, 61, 63, 64, 65, 72, 91, 98, 117, 124, 125, 126, 127, 128, 133, 152, 169, 189, 208, 215, 216, 217, 218, 224, 243, 250, 271, 279, 280, 296, 316, 331, 335, 341, 342, 343, 344, 351, 370, 386, 387, 397, 407, 432 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Sums of two integer cubes. - Charles R Greathouse IV, Mar 30 2022
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 86.
LINKS
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
EXAMPLE
7 = (2)^3 + (-1)^3.
MATHEMATICA
Union[Select[Sort[Flatten[Table[{j^3-i^3, j^3+i^3}, {i, 0, 20}, {j, i, 20}]]], #<20^3-19^3&]]
With[{nn=20}, Take[Union[Select[Flatten[{Total[#], #[[1]]-#[[2]]}&/@(Tuples[ Range[0, nn], 2]^3)], #>-1&]], 3*nn]] (* Harvey P. Dale, Jun 22 2014 *)
PROG
(PARI) is(n)=fordiv(n, d, my(L=(d^2-n/d)/3); if(denominator(L)==1 && issquare(d^2-4*L), return(1))); 0 \\ Charles R Greathouse IV, Jun 12 2012
(PARI) list(lim)={
my(v=List(), x3, t);
for(x=0, sqrtnint(lim\=1, 3),
x3=x^3;
for(y=0, min(sqrtnint(lim-x3, 3), x),
listput(v, x3+y^3)
)
);
for(x=2, t=sqrtint(lim\3),
x3=x^3;
for(y=sqrtnint(max(0, x3-lim-1), 3)+1, x-1,
listput(v, x3-y^3)
)
);
t=(t+1)^3-t^3;
if(t<=lim, listput(v, t));
Set(v);
} \\ Charles R Greathouse IV, Jun 12 2012, updated Jan 13 2022
(PARI) is(n)=#thue(thueinit(z^3+1), n) \\ Ralf Stephan, Oct 18 2013
(Haskell)
a045980 n = a045980_list !! (n-1)
a045980_list = 0 : filter f [1..] where
f x = g $ takeWhile ((<= 4 * x) . (^ 3)) $ a027750_row x where
g [] = False
g (d:ds) = r == 0 && a010052 (d ^ 2 - 4 * y) == 1 || g ds
where (y, r) = divMod (d ^ 2 - div x d) 3
-- Reinhard Zumkeller, Dec 20 2013
CROSSREFS
A004999 and A003325 are subsequences.
Sequence in context: A037455 A020675 A317303 * A104339 A199004 A168064
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified May 21 05:34 EDT 2024. Contains 372728 sequences. (Running on oeis4.)