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A003328 Numbers that are the sum of 5 positive cubes.

%I A003328 #37 Jun 14 2025 03:04:23
%S A003328 5,12,19,26,31,33,38,40,45,52,57,59,64,68,71,75,78,82,83,89,90,94,96,
%T A003328 97,101,108,109,115,116,120,127,129,131,134,135,136,138,143,145,146,
%U A003328 150,152,153,155,157,162,164,169,171,172,176,181,183,188,190,192,194,195,199
%N A003328 Numbers that are the sum of 5 positive cubes.
%C A003328 As the order of addition doesn't matter we can assume terms are in increasing order. - _David A. Corneth_, Aug 01 2020
%C A003328 It seems only a finite number N of positive integers are not in this sequence, and thus a(n) = n - N for all sufficiently large n. Is it true that 2243453, last term of A048927, is sufficiently large in that sense? - _M. F. Hasler_, Jan 04 2023
%H A003328 David A. Corneth, <a href="/A003328/b003328.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A003328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubicNumber.html">Cubic Number.</a>
%e A003328 From _David A. Corneth_, Aug 01 2020: (Start)
%e A003328 3084 is in the sequence as 3084 = 5^3 + 5^3 + 5^3 +  8^3 + 13^3.
%e A003328 4385 is in the sequence as 4385 = 4^3 + 4^3 + 9^3 + 11^3 + 13^3.
%e A003328 5426 is in the sequence as 5426 = 8^3 + 9^3 + 9^3 + 12^3 + 12^3. (End)
%o A003328 (PARI) select( {is_A003328(n,k=5,m=3,L=sqrtnint(abs(n-k+1),m))=if( n>k*L^m || n<k, 0, n<k*L^m, forstep(r=min(k-1,n\L^m),0,-1, self()(n-r*L^m,k-r,m,L-1) && return(1)), 1)}, [1..200]) \\ _M. F. Hasler_, Aug 02 2020
%o A003328 A003328_upto(N,k=5,m=3)=[i|i<-[1..#N=sum(n=1,sqrtnint(N,m),'x^n^m,O('x^N))^k], polcoef(N,i)] \\ _M. F. Hasler_, Aug 02 2020
%o A003328 (Python)
%o A003328 from collections import Counter
%o A003328 from itertools import combinations_with_replacement as combs_w_rep
%o A003328 def aupto(lim):
%o A003328   s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
%o A003328   s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_w_rep(s, 5)))
%o A003328   s2counts = Counter(s2)
%o A003328   return sorted(k for k in s2counts)
%o A003328 print(aupto(200)) # _Michael S. Branicky_, May 12 2021
%Y A003328 Cf. A003328, A048926, A048297.
%Y A003328 Cf. A057906 (Complement)
%Y A003328 Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
%Y A003328   A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
%K A003328 nonn,easy
%O A003328 1,1
%A A003328 _N. J. A. Sloane_