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%I #82 Oct 29 2023 01:46:02
%S 3,10,17,24,29,36,43,55,62,66,73,80,81,92,99,118,127,129,134,136,141,
%T 153,155,160,179,190,192,197,216,218,225,232,244,251,253,258,270,277,
%U 281,288,307,314,342,344,345,349,352,359,368,371,375,378,397,405,408,415,433,434
%N Numbers that are the sum of 3 positive cubes.
%C A119977 is a subsequence; if m is a term then there exists at least one k>0 such that m-k^3 is a term of A003325. - _Reinhard Zumkeller_, Jun 03 2006
%C A025456(a(n)) > 0. - _Reinhard Zumkeller_, Apr 23 2009
%C Davenport proved that a(n) << n^(54/47 + e) for every e > 0. - _Charles R Greathouse IV_, Mar 26 2012
%H K. D. Bajpai, <a href="/A003072/b003072.txt">Table of n, a(n) for n = 1..12955</a> (first 1000 terms from T. D. Noe)
%H H. Davenport, <a href="https://doi.org/10.1112/jlms/s1-25.4.339">Sums of three positive cubes</a>, J. London Math. Soc., 25 (1950), 339-343. Coll. Works III p. 999.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F {n: A025456(n) >0}. - _R. J. Mathar_, Jun 15 2018
%e a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes.
%e a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.
%p isA003072 := proc(n)
%p local x,y,z;
%p for x from 1 do
%p if 3*x^3 > n then
%p return false;
%p end if;
%p for y from x do
%p if x^3+2*y^3 > n then
%p break;
%p end if;
%p if isA000578(n-x^3-y^3) then
%p return true;
%p end if;
%p end do:
%p end do:
%p end proc:
%p for n from 1 to 1000 do
%p if isA003072(n) then
%p printf("%d,",n) ;
%p end if;
%p end do: # _R. J. Mathar_, Jan 23 2016
%t Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* _Jean-François Alcover_, Apr 29 2011 *)
%t With[{upto=500},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2,3]]]^3,3]],#<=upto&]] (* _Harvey P. Dale_, Oct 25 2021 *)
%o (PARI) sum(n=1,11,x^(n^3),O(x^1400))^3 /* Then [i|i<-[1..#%],polcoef(%,i)] gives the list of powers with nonzero coefficient. - _M. F. Hasler_, Aug 02 2020 */
%o (PARI) list(lim)=my(v=List(),k,t); lim\=1; for(x=1,sqrtnint(lim-2,3), for(y=1, min(sqrtnint(lim-x^3-1,3),x), k=x^3+y^3; for(z=1,min(sqrtnint(lim-k,3), y), listput(v, k+z^3)))); Set(v) \\ _Charles R Greathouse IV_, Sep 14 2015
%o (Haskell)
%o a003072 n = a003072_list !! (n-1)
%o a003072_list = filter c3 [1..] where
%o c3 x = any (== 1) $ map (a010057 . fromInteger) $
%o takeWhile (> 0) $ map (x -) $ a003325_list
%o -- _Reinhard Zumkeller_, Mar 24 2012
%Y Subsequence of A004825.
%Y Cf. A003325, A024981, A057904 (complement), A010057, A000578, A023042 (subsequence of cubes).
%Y Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers: 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,nice
%O 1,1
%A _N. J. A. Sloane_, _David W. Wilson_
%E Incorrect program removed by _David A. Corneth_, Aug 01 2020
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