A003331 - OEIS

%I #31 Dec 26 2021 20:43:59

%S 8,15,22,29,34,36,41,43,48,50,55,57,60,62,64,67,69,71,74,76,78,81,83,

%T 85,86,88,92,93,95,97,99,100,102,104,106,107,111,112,113,114,118,119,

%U 120,121,123,125,126,130,132,133,134,137,138,139,140,141,144,145,146,148,149

%N Numbers that are the sum of 8 positive cubes.

%C 620 is the largest among only 142 positive integers not in this sequence. This can be proved by induction. - _M. F. Hasler_, Aug 13 2020

%H David A. Corneth, <a href="/A003331/b003331.txt">Table of n, a(n) for n = 1..10000</a>

%H OEIS Wiki, <a href="/wiki/Index_to_Sums_of_like_powers">Index to sequences related to sums of like powers</a>.

%F a(n) = 142 + n for all n > 478. - _M. F. Hasler_, Aug 13 2020

%e From _David A. Corneth_, Aug 01 2020: (Start)

%e 1796 is in the sequence as 1796 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 7^3 + 7^3 + 9^3.

%e 2246 is in the sequence as 2246 = 2^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 7^3 + 11^3.

%e 3164 is in the sequence as 3164 = 5^3 + 5^3 + 6^3 + 6^3 + 8^3 + 8^3 + 9^3 + 9^3.(End)

%t Module[{upto=200,c},c=Floor[Surd[upto,3]];Select[Union[Total/@ Tuples[ Range[ c]^3,8]],#<=upto&]] (* _Harvey P. Dale_, Jan 11 2016 *)

%o (PARI) (A003331_upto(N, k=8, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ _M. F. Hasler_, Aug 02 2020

%o A003331(n)=if(n>478, n+142, n>329, n+141, A003331_upto(470)[n]) \\ _M. F. Hasler_, Aug 13 2020

%o (Python)

%o from itertools import combinations_with_replacement as mc

%o def aupto(lim):

%o     cbs = (i**3 for i in range(1, int((lim-7)**(1/3))+2))

%o     return sorted(set(k for k in (sum(c) for c in mc(cbs, 8)) if k <= lim))

%o print(aupto(150)) # _Michael S. Branicky_, Aug 15 2021

%Y Other sequences of numbers that are the sum of x nonzero y-th powers:

%Y   A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),

%Y   A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),

%Y   A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),

%Y   A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),

%Y   A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),

%Y   A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_