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%I #12 Jul 23 2023 13:42:13
%S 0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,7,
%T 7,8,8,8,9,10,10,10,10,10,10,10,10,12,12,12,13,13,14,15,16,16,16,17,
%U 17,19,19,19,19,20,20,20,21,23,24,24,24,25,25,25,25
%N Number of sums i^3 + j^3 that occur more than once for 1<=i<=j<=n.
%H Seiichi Manyama, <a href="/A061798/b061798.txt">Table of n, a(n) for n = 1..1000</a>
%e If the {s+t} sums are generated by adding 2 terms of an S set consisting of n different entries, then at least 1, at most n(n+1)/2=A000217(n) distinct values can be obtained. The set of first n cubes gives results falling between these two extremes. E.g. S={1,8,27,...,2744,3375} provides 119 different sums of two, not necessarily different cubes:{2,9,....,6750}. Only a single sum occurs more than once: 1729(Ramanujan): 1729=1+1728=729+1000. Therefore a(15)=C[15,2]+15-119=120-119=1.
%t f[x_] := x^3 t0=Table[Length[Union[Flatten[Table[f[u]+f[w], {w, 1, m}, {u, 1, m}]]]], {m, 1, 75}] t1=Table[(w*(w+1)/2)-Part[t0, w], {w, 1, 75}]
%Y Cf. A000217, A061791.
%K nonn
%O 1,16
%A _Labos Elemer_, Jun 22 2001
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