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A360619 a(n) > n is the smallest integer such that there exist integers n < c < d satisfying n^3 + a(n)^3 = c^3 + d^3.

%I A360619 #38 Jan 08 2025 11:38:32
%S A360619 12,16,36,32,60,48,84,53,34,27,93,40,156,112,80,106,39,68,228,54,238,
%T A360619 176,94,80,167,156,102,224,99,67,246,166,279,78,98,120,174,304,468,
%U A360619 108,319,69,516,352,170,188,97,160,282,96,82,312,550,204,113,371,180,198,708,134,600
%N A360619 a(n) > n is the smallest integer such that there exist integers n < c < d satisfying n^3 + a(n)^3 = c^3 + d^3.
%C A360619 Since the identity n^3 + (12n)^3 = (9n)^3 + (10n)^3 holds, n < a(n) <= 12n.
%H A360619 Jean-François Alcover, <a href="/A360619/b360619.txt">Table of n, a(n) for n = 1..200</a> (terms 1..61 from Giedrius Alkauskas).
%H A360619 Giedrius Alkauskas, <a href="https://web.vu.lt/mif/g.alkauskas/math/cubes.pdf">On the sequence representing narrowest intervals containing two pairs of integers with equal cube sums</a>.
%e A360619 For n = 11, a(11) = 93, since, first, 11^3 + 93^3 = 30^3 + 92^3. Second, for any integral y in the range [12, 92] there does not exist c, d, 11 < c < d < y, satisfying 11^3 + y^3 = c^3 + d^3.
%p A360619 a :=proc(n::integer) local found::boolean; local N, SQ, i;
%p A360619 found:=false; N:=n+1; SQ:={};
%p A360619 while not found do SQ:=SQ union {N^3}; N:=N+1;
%p A360619 for i from n+1 to N-1 do if evalb(N^3+n^3-i^3 in SQ) then
%p A360619 found:=true; end if; end do; end do; N end proc;
%t A360619 a[n_] := a[n] = Module[{found, m, SQ, i}, found = False; m = n+1; SQ = {}; While[!found, SQ = SQ ~Union~ {m^3}; m = m+1; For[i = n+1, i <= m-1, i++, If[MemberQ[SQ, m^3+n^3-i^3], found = True]]]; m];
%t A360619 Table[Print[n, " ", a[n]]; a[n], {n, 1, 200}] (* _Jean-François Alcover_, Feb 27 2023, after _Giedrius Alkauskas_'s Maple code *)
%Y A360619 Cf. A360427, A001235.
%K A360619 nonn
%O A360619 1,1
%A A360619 _Giedrius Alkauskas_, Feb 14 2023