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%I A003072 #82 Oct 29 2023 01:46:02
%S A003072 3,10,17,24,29,36,43,55,62,66,73,80,81,92,99,118,127,129,134,136,141,
%T A003072 153,155,160,179,190,192,197,216,218,225,232,244,251,253,258,270,277,
%U A003072 281,288,307,314,342,344,345,349,352,359,368,371,375,378,397,405,408,415,433,434
%N A003072 Numbers that are the sum of 3 positive cubes.
%C A003072 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 A003072 A025456(a(n)) > 0. - _Reinhard Zumkeller_, Apr 23 2009
%C A003072 Davenport proved that a(n) << n^(54/47 + e) for every e > 0. - _Charles R Greathouse IV_, Mar 26 2012
%H A003072 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 A003072 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 A003072 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>
%H A003072 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A003072 {n: A025456(n) >0}. - _R. J. Mathar_, Jun 15 2018
%e A003072 a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes.
%e A003072 a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.
%p A003072 isA003072 := proc(n)
%p A003072     local x,y,z;
%p A003072     for x from 1 do
%p A003072         if 3*x^3 > n then
%p A003072             return false;
%p A003072         end if;
%p A003072         for y from x do
%p A003072             if x^3+2*y^3 > n then
%p A003072                 break;
%p A003072             end if;
%p A003072             if isA000578(n-x^3-y^3) then
%p A003072                 return true;
%p A003072             end if;
%p A003072         end do:
%p A003072     end do:
%p A003072 end proc:
%p A003072 for n from 1 to 1000 do
%p A003072     if isA003072(n) then
%p A003072         printf("%d,",n) ;
%p A003072     end if;
%p A003072 end do: # _R. J. Mathar_, Jan 23 2016
%t A003072 Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* _Jean-François Alcover_, Apr 29 2011 *)
%t A003072 With[{upto=500},Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2,3]]]^3,3]],#<=upto&]] (* _Harvey P. Dale_, Oct 25 2021 *)
%o A003072 (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 A003072 (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 A003072 (Haskell)
%o A003072 a003072 n = a003072_list !! (n-1)
%o A003072 a003072_list = filter c3 [1..] where
%o A003072    c3 x = any (== 1) $ map (a010057 . fromInteger) $
%o A003072                        takeWhile (> 0) $ map (x -) $ a003325_list
%o A003072 -- _Reinhard Zumkeller_, Mar 24 2012
%Y A003072 Subsequence of A004825.
%Y A003072 Cf. A003325, A024981, A057904 (complement), A010057, A000578, A023042 (subsequence of cubes).
%Y A003072 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 A003072 nonn,easy,nice
%O A003072 1,1
%A A003072 _N. J. A. Sloane_, _David W. Wilson_
%E A003072 Incorrect program removed by _David A. Corneth_, Aug 01 2020

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