

A217843


Numbers which are the sum of one or more consecutive nonnegative cubes.


26



0, 1, 8, 9, 27, 35, 36, 64, 91, 99, 100, 125, 189, 216, 224, 225, 341, 343, 405, 432, 440, 441, 512, 559, 684, 729, 748, 775, 783, 784, 855, 1000, 1071, 1196, 1241, 1260, 1287, 1295, 1296, 1331, 1584, 1728, 1729, 1800, 1925, 1989, 2016, 2024, 2025, 2197
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OFFSET

1,3


COMMENTS

a(n) can always be expressed as the difference of the squares of two triangular numbers (A000217).
a(n) is also the product of two nonnegative integers whose sum and difference are both promic.
See example and formula sections for details. (End)


LINKS



FORMULA

a(n) is of the form [x*(x+2*k+1)*(x*(x+2*k+1)+2*k*(k+1))]/4, sum of n consecutive cubes starting from (k+1)^3.  Lamine Ngom, Apr 15 2021


EXAMPLE

Arrange the positive terms in a triangle as follows:
n\k  1 2 3 4 5 6 7
+
0  1;
1  8, 9;
2  27, 35, 36;
3  64, 91, 99, 100;
4  125, 189, 216, 224, 225;
5  216, 341, 405, 432, 440, 441;
6  343, 559, 684, 748, 775, 783, 784;
The difference of the squares of two consecutive triangular numbers (A000217) is a cube (A000578).
Column 2: sums of 2 consecutive cubes (A027602).
Column 3: sums of 3 consecutive cubes (A027603).
etc.
Column k: sums of k consecutive cubes.
T(n,n) = A000217(n)^2 (main diagonal).
Now rectangularize this triangle as follows:
n\k  1 2 3 4 5 6 ...
+
0  1, 9, 36, 100, 225, 441, ...
1  8, 35, 99, 224, 440, 783, ...
2  27, 91, 216, 432, 775, 1287, ...
3  64, 189, 405, 748, 1260, 1989, ...
4  125, 341, 684, 1196, 1925, 2925, ...
5  216, 559, 1071, 1800, 2800, 4131, ...
6  343, 855, 1584, 2584, 3915, 5643, ...
The general form of terms is:
For k = 1, the sequence provides all cubes: T(n,1) = A000578(k).
For k = 2, T(n,2) = A005898(k), centered cube numbers, sum of two consecutive cubes.
For k = 3, T(n,3) = A027602(k), sum of three consecutive cubes.
For k = 4, T(n,4) = A027603(k), sum of four consecutive cubes.
For k = 5, T(n,5) = A027604(k), sum of five consecutive cubes.
T(n,n) = A116149(n), sum of n consecutive cubes after n^3 (main diagonal).
For n = 0, we obtain the subsequence T(0,k) = A000217(n)^2, product of two numbers whose difference is 0*1 (promic) and sum is promic too.
For n = 1, we obtain the subsequence T(1,k) = A168566(x), product of two numbers whose difference is 1*2 (promic) and sum is promic too.
For n = 2, we obtain the subsequence T(2,k) = product of two numbers whose difference is 2*3 (promic) and sum is promic too.
etc.
For n = x, we obtain the subsequence formed by products of two numbers whose difference is the promic x*(x+1) and sum is promic too.
Consequently, if m is in the sequence, then m can be expressed as the product of two nonnegative integers whose sum and difference are both promic. (End)


MATHEMATICA

nMax = 3000; t = {0}; Do[k = n; s = 0; While[s = s + k^3; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/3)}]; t = Union[t]


PROG

(Haskell)
import Data.Set (singleton, deleteFindMin, insert, Set)
a217843 n = a217843_list !! (n1)
a217843_list = f (singleton (0, (0, 0))) (1) where
f s z = if y /= z then y : f s'' y else f s'' y
where s'' = (insert (y', (i, j')) $
insert (y'  i ^ 3 , (i + 1, j')) s')
y' = y + j' ^ 3; j' = j + 1
((y, (i, j)), s') = deleteFindMin s
(PARI) lista(nn) = {my(list = List([0])); for (i=1, nn, my(s = 0); forstep(j=i, 1, 1, s += j^3; if (s > nn^3, break); listput(list, s); ); ); Set(list); } \\ Michel Marcus, Nov 13 2020


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



