A054205 - OEIS

%I #21 Sep 02 2023 17:00:18

%S 5,95,216,566,728,804,1625,2044,9393,11387,13503,14014,19520,22874,

%T 27545,34182,38368,39388,39878,50953,69977,76200,98494,107976,141750,

%U 143424,230094,251263,466655,521016,572886,616712,724625,1211267,1346927,1383317,1810998,1883791,1985794,1999374,2015494,2066496,2174526

%N Consider all integer triples (i,j,k), j >= k > 0, with binomial(i+2,3) = j^3 + k^3, ordered by increasing i; sequence gives i values.

%C The tetrahedral number A000292(a(n)-1) is the sum of two positive cubes. The j values are in A054206 and k values in A054207.

%C A sum of two cubes can only have 25 distinct values (mod 63). In a search, this means only 75 values for i (mod 189) need to be evaluated. - _Bert Dobbelaere_, Jan 13 2019

%H Bert Dobbelaere, <a href="/A054205/b054205.txt">Table of n, a(n) for n = 1..53</a>

%e binomial(5+2, 3) = 35 = 2^3 + 3^3;

%e binomial(95+2, 3) = 147440 = 49^3 + 31^3.

%t lst = {}; Do[ b = Binomial[i + 2, 3]; j = Floor[b^(1/3)]; lmt = Ceiling[j/2]; While[ k = (b - j^3)^(1/3); j > lmt && !IntegerQ[k], j-- ]; If[j != lmt, Print[{i, j, k}]; AppendTo[lst, {i, j, k}]], {i, 2, 30000}]; First /@ lst (* _Robert G. Wilson v_, Jan 15 2007 *)

%Y Cf. A054206, A054207, A000292.

%K nice,nonn

%O 1,1

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 28 2000

%E More terms from _Martin Fuller_, Nov 27 2006

%E Terms a(30)-a(36) from _Jon E. Schoenfield_, Jan 14 2009

%E Offset corrected by _N. J. A. Sloane_, Jan 14 2009

%E Terms a(37)-a(43) from _Jon E. Schoenfield_, Aug 30 2013