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.

12, 16, 36, 32, 60, 48, 84, 53, 34, 27, 93, 40, 156, 112, 80, 106, 39, 68, 228, 54, 238, 176, 94, 80, 167, 156, 102, 224, 99, 67, 246, 166, 279, 78, 98, 120, 174, 304, 468, 108, 319, 69, 516, 352, 170, 188, 97, 160, 282, 96, 82, 312, 550, 204, 113, 371, 180, 198, 708, 134, 600

Offset

1,1

Comments

Since the identity n^3 + (12n)^3 = (9n)^3 + (10n)^3 holds, n < a(n) <= 12n.

Links

Jean-François Alcover, Table of n, a(n) for n = 1..200 (terms 1..61 from Giedrius Alkauskas).

Giedrius Alkauskas, Euler's cubic equation and narrowest interval [a,b] containing a quadruple a<c<d<b such that a^3+b^3=c^3+d^3.

Example

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.

Maple

a :=proc(n::integer) local found::boolean; local N, SQ, i;
found:=false; N:=n+1; SQ:={};
while not found do SQ:=SQ union {N^3}; N:=N+1;
for i from n+1 to N-1 do if evalb(N^3+n^3-i^3 in SQ) then
found:=true; end if; end do; end do; N end proc;

Mathematica

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];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 200}] (* Jean-François Alcover, Feb 27 2023, after Giedrius Alkauskas's Maple code *)

Crossrefs

Cf. A360427, A001235.

Sequence in context: A086980 A256934 A343153 * A058203 A341297 A323978

Adjacent sequences: A360616 A360617 A360618 * A360620 A360621 A360622

Keyword

nonn

Author

Giedrius Alkauskas, Feb 14 2023

Status

approved