A024670 Numbers that are sums of 2 distinct positive cubes.

9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 513, 520, 539, 559, 576, 637, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1512, 1547, 1674

Offset

1,1

Comments

This sequence contains no primes since x^3+y^3=(x^2-x*y+y^2)*(x+y). - M. F. Hasler, Apr 12 2008

There are no terms == 3, 4, 5 or 6 mod 9. - Robert Israel, Oct 07 2014

a(n) mod 2: {1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,1,0, ...} - Daniel Forgues, Sep 27 2018

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..902 from M. F. Hasler)

Index to sequences related to sums of cubes

Example

9 is in the sequence since 2^3 + 1^3 = 9.
35 is in the sequence since 3^3 + 2^3 = 35.

Maple

N:= 10000: # to get all terms <= N
S:= select(`<=`, {seq(seq(i^3 + j^3, j = 1 .. i-1), i = 2 .. floor(N^(1/3)))}, N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(S, list));
# Robert Israel, Oct 07 2014

Mathematica

lst={}; Do[Do[x=a^3; Do[y=b^3; If[x+y==n, AppendTo[lst, n]], {b, Floor[(n-x)^(1/3)], a+1, -1}], {a, Floor[n^(1/3)], 1, -1}], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Select[Range@ 1700, Total@ Boole@ Map[And[! MemberQ[#, 0], UnsameQ @@ #] &, PowersRepresentations[#, 2, 3]] > 0 &] (* Michael De Vlieger, May 13 2017 *)

Prog

(PARI) isA024670(n)=for( i=ceil(sqrtn( n\2+1, 3)), sqrtn(n-.5, 3), isA000578(n-i^3) & return(1)) /* One could also use "for( i=2, sqrtn( n\2-1, 3), ...)" but this is much slower since there are less cubes in [n/2, n] than in [1, n/2]. Replacing the -1 here by +.5 would yield A003325, allowing for a(n)=x^3+x^3. Replacing -1 by 0 may miss some a(n) of this form due to rounding errors. - M. F. Hasler, Apr 12 2008 */
(Python)
from itertools import count, takewhile
def aupto(limit):
    cbs = list(takewhile(lambda x: x <= limit, (i**3 for i in count(1))))
    sms = set(c+d for i, c in enumerate(cbs) for d in cbs[i+1:])
    return sorted(s for s in sms if s <= limit)
print(aupto(1674)) # Michael S. Branicky, Sep 28 2021

Crossrefs

See also: Sums of 2 positive cubes (not necessarily distinct): A003325. Sums of 3 distinct positive cubes: A024975. Sums of distinct positive cubes: A003997. Sums of 2 distinct nonnegative cubes: A114090. Sums of 2 nonnegative cubes: A004999. Sums of 2 distinct positive squares: A004431. Cubes: A000578.

Sequence in context: A127629 A334185 A267686 * A141805 A256497 A124360

Adjacent sequences: A024667 A024668 A024669 * A024671 A024672 A024673

Keyword

nonn

Author

Clark Kimberling

Extensions

Name edited by Zak Seidov, May 31 2011

Status

approved