

A024670


Numbers that are sums of 2 distinct positive cubes.


39



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
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OFFSET

1,1


COMMENTS

This sequence contains no primes since x^3+y^3=(x^2x*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



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 .. i1), i = 2 .. floor(N^(1/3)))}, N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(S, list));


MATHEMATICA

lst={}; Do[Do[x=a^3; Do[y=b^3; If[x+y==n, AppendTo[lst, n]], {b, Floor[(nx)^(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(ni^3) & return(1)) /* One could also use "for( i=2, sqrtn( n\21, 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)


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.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



