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A024670
Numbers that are sums of 2 distinct positive cubes.
42
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
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.
Cf. A373971 (characteristic function).
Indices of nonzero terms in A025468.
Sequence in context: A127629 A334185 A267686 * A141805 A256497 A124360
KEYWORD
nonn
EXTENSIONS
Name edited by Zak Seidov, May 31 2011
STATUS
approved