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%I #35 Dec 01 2022 22:12:33
%S 0,1,2,8,9,16,27,28,35,54,64,65,72,91,125,126,128,133,152,189,216,217,
%T 224,243,250,280,341,343,344,351,370,407,432,468,512,513,520,539,559,
%U 576,637,686,728,729,730,737,756,793,854,855,945,1000,1001
%N Sums of two nonnegative cubes.
%H T. D. Noe, <a href="/A004999/b004999.txt">Table of n, a(n) for n = 1..1000</a>
%H Kevin A. Broughan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Broughan/broughan25.html">Characterizing the sum of two cubes</a>, J. Integer Seqs., Vol. 6, 2003.
%H Samuel S. Wagstaff, Jr., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Wagstaff/wagstaff8.html">Equal Sums of Two Distinct Like Powers</a>, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%t Union[(#[[1]]^3+#[[2]]^3)&/@Tuples[Range[0,20],{2}]] (* _Harvey P. Dale_, Dec 04 2010 *)
%o (PARI) is(n)=my(k1=ceil((n-1/2)^(1/3)), k2=floor((4*n+1/2)^(1/3)), L); fordiv(n,d,if(d>=k1 && d<=k2 && denominator(L=(d^2-n/d)/3)==1 && issquare(d^2-4*L), return(1))); 0
%o list(lim)=my(v=List());for(x=0,(lim+.5)^(1/3),for(y=0,min(x,(lim-x^3)^(1/3)),listput(v,x^3+y^3))); vecsort(Vec(v),,8) \\ _Charles R Greathouse IV_, Jun 12 2012
%o (PARI) is(n)=my(L=sqrtnint(n-1,3)+1,U=sqrtnint(4*n,3));fordiv(n,m,if(L<=m&m<=U,my(ell=(m^2-n/m)/3);if(denominator(ell)==1&&issquare(m^2-4*ell),return(1))));0 \\ _Charles R Greathouse IV_, Apr 16 2013
%o (PARI) T=thueinit('z^3+1);
%o is(n)=n==0 || #select(v->min(v[1],v[2])>=0, thue(T,n))>0 \\ _Charles R Greathouse IV_, Nov 29 2014
%o (Haskell)
%o a004999 n = a004999_list !! (n-1)
%o a004999_list = filter c2 [1..] where
%o c2 x = any (== 1) $ map (a010057 . fromInteger) $
%o takeWhile (>= 0) $ map (x -) $ tail a000578_list
%o -- _Reinhard Zumkeller_, Dec 20 2013
%Y Subsequence of A045980; A003325 is a subsequence.
%Y Cf. A000578, A004825, A010057
%K nonn,easy,nice
%O 1,3
%A _N. J. A. Sloane_, _Steven Finch_
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