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%I #33 Jan 06 2023 20:45:26
%S 5,12,19,26,31,33,38,40,45,52,57,59,64,68,71,75,78,82,83,89,90,94,96,
%T 97,101,108,109,115,116,120,127,129,131,134,135,136,138,143,145,146,
%U 150,152,153,155,157,162,164,169,171,172,176,181,183,188,190,192,194,195,199
%N Numbers that are the sum of 5 positive cubes.
%C As the order of addition doesn't matter we can assume terms are in increasing order. - _David A. Corneth_, Aug 01 2020
%C 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number.</a>
%e From _David A. Corneth_, Aug 01 2020: (Start)
%e 3084 is in the sequence as 3084 = 5^3 + 5^3 + 5^3 + 8^3 + 13^3.
%e 4385 is in the sequence as 4385 = 4^3 + 4^3 + 9^3 + 11^3 + 13^3.
%e 5426 is in the sequence as 5426 = 8^3 + 9^3 + 9^3 + 12^3 + 12^3. (End)
%o (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_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 (Python)
%o from collections import Counter
%o from itertools import combinations_with_replacement as combs_w_rep
%o def aupto(lim):
%o s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
%o s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_w_rep(s, 5)))
%o s2counts = Counter(s2)
%o return sorted(k for k in s2counts)
%o print(aupto(200)) # _Michael S. Branicky_, May 12 2021
%Y Cf. A003328, A048926, A048297.
%Y Cf. A057906 (Complement)
%Y Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
%Y 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 nonn,easy
%O 1,1
%A _N. J. A. Sloane_
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