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A202679 Numbers that are sums of two coprime positive cubes. 7
2, 9, 28, 35, 65, 91, 126, 133, 152, 189, 217, 341, 344, 351, 370, 407, 468, 513, 539, 559, 637, 730, 737, 793, 854, 855, 1001, 1027, 1072, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1547, 1674, 1729, 1843, 1853, 2060, 2071, 2198, 2205, 2224, 2261, 2322, 2331, 2413 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Not a subsequence of A020898: non-cubefree members of this sequence include 152, 189, 344, 351, 513, 1072. - Robert Israel, Mar 16 2016
LINKS
Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
R. C. Baker, Sums of two relatively prime cubes, Acta Arithmetica 129(2007), 103-146.
Kevin A. Broughan, A computational approach to characterizing the sum of two cubes, Hamilton: University of Waikato, 2001, p. 9.
P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), pp. 134-139. [alternate link]
P. Erdős, On the integers of the form x^k + y^k, J. London Math. Soc. 14 (1939), pp. 250-254.
FORMULA
Erdős & Mahler shows that a(n) < kn^(3/2) for some k. Erdős later gives an elementary proof. - Charles R Greathouse IV, Dec 05 2012
EXAMPLE
28 is in the sequence since 1^3 + 3^3 = 28 and (1, 3) = 1.
MAPLE
N:= 10000: # to get all terms <= N
S:= {2, seq(seq(x^3 + y^3, y = select(t -> igcd(t, x)=1, [$x+1 .. floor((N - x^3)^(1/3))])), x = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S, list)); # Robert Israel, Mar 15 2016
MATHEMATICA
nn = 2500; Union[Flatten[Table[If[CoprimeQ[x, y] == True, x^3 + y^3, {}], {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]
Select[Range@ 2500, Length[PowersRepresentations[#, 2, 3] /. {{0, _} -> Nothing, {a_, b_} /; ! CoprimeQ[a, b] -> Nothing}] > 0 &] (* Michael De Vlieger, Mar 15 2016 *)
PROG
(PARI) is(n)=for(k=1, (n\2+.5)^(1/3), if(gcd(k, n)==1&&ispower(n-k^3, 3), return(1))); 0 \\ Charles R Greathouse IV, Apr 13 2012
(PARI) list(lim)=my(v=List()); forstep(x=1, lim^(1/3), 2, forstep(y=2, (lim-x^3+.5)^(1/3), 2, if(gcd(x, y)==1, listput(v, x^3+y^3))); forstep(y=1, min((lim-x^3+.5)^(1/3), x), 2, if(gcd(x, y)==1, listput(v, x^3+y^3)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Dec 05 2012
CROSSREFS
Subsequence of A003325.
Sequence in context: A357721 A155472 A100293 * A340049 A256467 A303373
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 18 02:04 EDT 2024. Contains 372615 sequences. (Running on oeis4.)