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Sums of two or more consecutive positive cubes.
3

%I #29 Dec 25 2015 15:01:00

%S 9,35,36,91,99,100,189,216,224,225,341,405,432,440,441,559,684,748,

%T 775,783,784,855,1071,1196,1241,1260,1287,1295,1296,1584,1729,1800,

%U 1925,1989,2016,2024,2025,2241,2331,2584,2800,2925,2989,3016,3024,3025,3059,3060

%N Sums of two or more consecutive positive cubes.

%C All numbers of the form A000537(b) - A000537(a) for 0 <= a <= b-2.

%C A217843 minus (A000578 minus A131643).

%C n is in the sequence iff n = s*t where (s+t)/2 = A000217(u) and (s-t)/2 = A000217(v) with u-v >= 2.

%C If a(k(n)) = A000537(n+1), k(n) >= A000217(n) for n > 0. - _Altug Alkan_, Dec 07 2015

%C See A062682 for sums of two or more consecutive positive cubes in more than one way. - _Reinhard Zumkeller_, Dec 16 2015

%H Robert Israel, <a href="/A265377/b265377.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 1^3 + 2^3 = 9.

%e a(2) = 2^3 + 3^3 = 35.

%e a(3) = 1^3 + 2^3 + 3^3 = 36.

%p amin:= proc(b,N) local r;

%p r:= b^2*(b+1)^2 - 4*N; if r > 0 then iroot(r,4) else 1 fi

%p end proc:

%p A265377:= proc(N) # to get all terms <= N

%p local a,b;

%p sort(convert(select(`<=`,{seq(seq(b^2*(b+1)^2/4 - a^2*(a-1)^2/4,

%p a = amin(b,N) .. b-1), b=2..1+iroot(floor(N/2),3))},N),list))

%p end proc:

%p A265377(10000);

%t With[{nn=12},Select[Sort[Flatten[Table[Total/@Partition[Range[nn]^3,n,1],{n,2,nn}]]],#<=((nn(nn+1))/2)^3&]] (* _Harvey P. Dale_, Dec 25 2015 *)

%o (Haskell)

%o import Data.Set (singleton, deleteFindMin, insert, Set)

%o a265377 n = a265377_list !! (n-1)

%o a265377_list = f (singleton (1 + 2^3, (1, 2))) (-1) where

%o f s z = if y /= z then y : f s'' y else f s'' y

%o where s'' = (insert (y', (i, j')) $

%o insert (y' - i ^ 3 , (i + 1, j')) s')

%o y' = y + j' ^ 3; j' = j + 1

%o ((y, (i, j)), s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, Dec 17 2015

%Y Subset of A217843.

%Y Cf. A000217, A000537, A000578, A131643.

%Y Cf. A062682 (subsequence).

%K nonn

%O 1,1

%A _Robert Israel_, Dec 07 2015