A003072 -id:A003072 - cp's OEIS frontend

A069137 Numbers which are sums of neither 1, 2, 3, 4, 5 or 6 nonnegative cubes.

7, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356, 363, 364

Offset: 1

N. J. A. Sloane, Apr 08 2002; edited Sep 15 2006

nonn

Sequence is conjectured to be finite.

			Numbers which need at least seven terms to represent them as a sum of positive cubes: 14=8+1+1+1+1+1+1.

Bohman, Jan and Froberg, Carl-Erik; Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.

T. D. Noe, Table of n, a(n) for n=1..138
Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau; appendix by I. Gusti Putu Purnaba, 7373170279850, Math. Comp. 69 (2000), 421-439.
Index entries for sequences related to sums of cubes

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A000578, A085334.

Natural numbers remaining if union of A003325, A003072, A003327, A003328, A003329 and A000578 sets were deleted. Remark: this sequence itself does not include cubes, in contrast to A085334.

A226955 Number of representations of n! as a sum of 3 positive cubes.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 4, 0, 2, 2, 6, 2, 11, 2, 13, 20, 24, 9

Offset: 1

Zak Seidov, Jun 26 2013

Conjecture: For any t >= 0, there are only finitely many values of n such that a(n) = t. - Altug Alkan, Aug 16 2020

			n = 12, n! = 479001600 = x^3 + y^3 + z^3 with {x,y,z} = {35,309,766}, {47,214,777}, {60,486,714}, {240,504,696}; 4 solutions, hence a(12) = 4;
n = 16, n! = x^3 + y^3 + z^3 with {x,y,z} = {7644,21192,22212}, {8240,8400,27040}, {10980,15288,25212}, {11648,18016,23808}, {12096,19968,22368}, {13030,18330,23240}; 6 solutions, hence a(16) = 6.
From _Chai Wah Wu_, May 21 2017: (Start)
n = 22, n! = x^3 + y^3 + z^3 with (x,y,z) = (286272, 8168832, 8334144), (443100, 4806340, 10042760), (663040, 7882560, 8590400), (720720, 5343408, 9902592), (757890, 8108100, 8389710), (854812, 2888886, 10320506), (861120, 3584160, 10251360), (1025640, 2784600, 10326960), (1266408, 4510296, 10099728), (1443806, 7114569, 9129295), (1792350, 6013602, 9657648), (1814400, 3689280, 10221120), (1871415, 4292190, 10126305), (1926720, 5685120, 9771840), (2419200, 7506240, 8823360), (2517424, 7223832, 9008552), (2779200, 3144960, 10232640), (2870532, 6957468, 9140040), (3021408, 4549080, 10007592), (3244410, 7888800, 8429190), (3776535, 6384105, 9321480), (5083936, 5242592, 9467136), (5681592, 7233408, 8253000), (6391665, 6719895, 8239770)
n = 23, n! = x^3 + y^3 + z^3 with (x,y,z) = (136080, 8250480, 29352960), (5369910, 6098890, 29422400), (5766592, 18082176, 27029696), (6151320, 19606860, 26247060), (7572485, 23185155, 23485930), (8255520, 10856160, 28848960), (8678304, 19104696, 26316360), (11959740, 19850400, 25365060), (13799880, 22091640, 23172240)
(End)

Lars Blomberg, Table of n, a(n) for n = 1..21, with solutions

Cf. A003072 (numbers that are the sum of 3 positive cubes), A267414.

a(n) = A025456(n!). - Charles R Greathouse IV, Oct 27 2013

a(17)-a(18) from Giovanni Resta, Jun 26 2013
a(4) corrected and a(19)-a(21) from Lars Blomberg, Sep 07 2013
a(22)-a(23) from Chai Wah Wu, May 21 2017

A084355 Least number of positive cubes needed to represent n!.

Original entry on oeis.org

1, 1, 2, 6, 3, 5, 5, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Hugo Pfoertner, Jun 22 2003

nonn

			a(4)=3 because 4!=24=2^3+2^3+2^3.
a(0)=1 because 0!=1=1^3.
a(1)=1 because 1!=1=1^3.
a(2)=2 because 2!=2=1^3+1^3.
a(3)=6 because 3!=6=1^3+1^3+1^3+1^3+1^3+1^3.
a(4)=3 because 4!=24=2^3+2^3+2^3.
a(5)=5 because 5!=120=1^3+3^3+3^3+4^3+1^3.
a(6)=5 because 6!=720=4^3+6^3+6^3+6^3+2^3.
a(7)=4 because 7!=5040=1^3+5^3+17^3+1^3.
a(8)=4 because 8!=40320=2^3+10^3+34^3+2^3.
a(9)=3 because 9!=362880=52^3+56^3+36^3.
a(10)=3 because 10!=3628800=96^3+140^3+4^3.
a(11)=3 because 11!=39916800=222^3+303^3+105^3.
a(12)=3 because 12!=479001600=214^3+777^3+47^3.
a(13)=4 because 13!=6227020800=106^3+255^3+1838^3+33^3.
a(14)=3 because 14!=87178291200=1344^3+4392^3+312^3.
a(15)=3 because 15!=1307674368000=2040^3+10908^3+1092^3.
a(16)=3 because 16!=20922789888000=8400^3+27040^3+8240^3.
a(17)=3 because 17!=355687428096000=22848^3+69984^3+9984^3.
a(18)=3 because 18!=6402373705728000=54060^3+184080^3+18900^3.
From _Donovan Johnson_, May 17 2010: (Start)
a(19)=3 because 19!=121645100408832000=131040^3+331200^3+436320^3.
a(20)=3 because 20!=2432902008176640000=87490^3+1034430^3+1098440^3.
(End)

Cf. A000142, A002376, A002325, A003072, A003327, A003328, A003329.

a(n,up,dw,k)=local(i,m);if(k==1,if(n==round(sqrtn(n,3))^3,return(1),return(-1)),forstep(i=up,dw,-1,m=n-i^3;if(a(m,min(i,floor(sqrtn(m,3))),ceil(sqrtn(m/(k-1),3)),k-1)==1,return(1)))) for(n=0,18,for(k=1,9,if(a(n!,floor(sqrtn(n!,3)),ceil(sqrtn(n!/k,3)),k)==1,print1(k", ");break))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 01 2007

a(n)=A002376(n!).

More terms from David W. Wilson, Jun 23 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 01 2007
a(19)-a(20) from Donovan Johnson, May 17 2010

A085334 Numbers which are neither sums of 2,3,4,5 or that of 6 nonnegative cubes.

Original entry on oeis.org

1, 7, 8, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 125, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356

Offset: 1

Labos Elemer, Jul 07 2003

nonn

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A069137.

Remaining set of all natural numbers if union of A003325, A003072, A003327, A003328 and A003329 sets were deleted.Remark: this sequence includes those cubes too, which are sums of 7 or more cubes.

A085336 Numbers which are sums of two and also sums of three positive cubes.

Original entry on oeis.org

344, 855, 1072, 1674, 2752, 3402, 3500, 3744, 4439, 4941, 5256, 6244, 6840, 6867, 6984, 8576, 9288, 9604, 9728, 10261, 10656, 10745, 10773, 10989, 13357, 13392, 14167, 14364, 15093, 16480, 17080, 17603, 17920, 18305, 18369, 18648, 20026

Offset: 1

Labos Elemer, Jul 07 2003

nonn

			1072 is a term because 1072 = 729 + 343 = 1000 + 64 + 8.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A003072, A003325.

Cases[{i = #, IntegerPartitions[i, {#}, Range@CubeRoot@i^3] & /@ {2, 3}} & /@Range@30000, {a_, {?(# != {} &) ..}} :> a] (* _Hans Rudolf Widmer, Dec 08 2024 *)

A104054 Numbers which are the sum of three positive cubes and divisible by 31.

Original entry on oeis.org

62, 155, 434, 496, 713, 775, 1054, 1240, 1333, 1457, 1581, 1674, 2232, 2325, 2604, 2883, 3224, 3410, 3472, 3503, 3689, 3782, 3968, 4061, 4185, 4402, 4929, 5394, 5580, 5704, 5921, 6200, 6572, 6758, 6913, 7750, 8091, 8370, 8432, 8494, 9083, 9827, 9920

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 02 2005

nonn

			a(1)=62 because 62=2^3+3^3+3^3.
a(2)=155 because 155=3^3+4^3+4^3.
a(3)=434 because 434=3^3+4^3+7^3......

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A003072.

cbQ[n_]:=Module[{prs=Select[PowersRepresentations[n,3,3], First[#]>0&]}, Length[ prs]>0]; Select[31Range[320],cbQ] (* Harvey P. Dale, Jul 27 2011 *)

A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.

Original entry on oeis.org

9, 66, 56, 108, 144, 192, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152

Offset: 2

Jonathan Vos Post, Sep 23 2006

nonn

Sequence begins with n = 2 because a(1) is undefined (sum of one positive cube cannot have exactly one prime factor, i.e., be prime).

			a(2) = least semiprime in A003325 = 9 = 3 * 3 = 1^3 + 2^3 = A085366(1).
a(3) = least 3-almost prime in A003072 = 66 = 2 * 3 * 11 = 1^3 + 1^3 + 4^3 = A003072(10).
a(4) = least 4-almost prime in A003327 = 56 = 2^3 * 7 = 1^3 + 1^3 + 3^3 + 3^3 = A003327(10).
a(5) = least 5-almost prime in A003328 = 108 = 2^2 * 3^3 = 4^3 + 3^3 + 2^3 + 2^3 + 1^3 = A003328(25).
a(6) = least 6-almost prime in A003329 = 144 = 2^4 * 3^2 = 5^3 + 2^3 + 2^3 + 1^3 + 1^3 + 1^3 = A003329(46).

Cf. A000578, A001222, A003072, A003325, A003327, A003328, A003329, A085366.

isSumcPosC := proc(n,c,minb)
        local nrt ;
        if c = 1 then nrt := iroot(n,3) ; if nrt^3 = n  and n>= minb then true; else false; end if;
        else for b from minb do if b^3 > n then return false; end if; if isSumcPosC(n-b^3,c-1,b) then return true; end if; end do: end if;
end proc:
A122733 := proc(n)
        for a from 1 do if numtheory[bigomega](a) = n then if isSumcPosC(a,n,1) then return a; end if; end if;
        end do:
end proc:
for n from 2 do print(A122733(n)) ; end do: # R. J. Mathar, Dec 22 2010

a(n) = Min{x = (c_1)^3 + (c_2)^3 + ... + (c_n)^3 such that omega(x) = A001222(x) = n}.

a(17) from Giovanni Resta, Jun 13 2016

a(18)-a(21) more terms from R. J. Mathar, Jan 31 2017

A271717 Integers k such that both k and k^3-1 are the sum of two positive cubes (see A003325).

Original entry on oeis.org

9, 11664, 36864, 38134, 345744, 1750329, 4782969, 20820969, 47775744, 65804544, 95004009, 150994944, 448084224, 733055625, 1093955625, 1416167424, 2197265625, 4318066944, 5194805625, 6198727824, 7169347584, 10771948944, 13013105625, 19591041024, 32427005625

Offset: 1

Altug Alkan, Apr 12 2016

nonn

Values of a^3 + b^3 such that (a^3 + b^3)^3 - 1 is of the form x^3 + y^3 where a, b, x, y > 0.
38134 = 2*23*829 is the first term that is nonsquare. What are the next square terms of this sequence?
n is a member of A007412 and n^3 is a member of A003072, obviously.

			9 is a term because 9 = 1^3 + 2^3 and 9^3 - 1 = 6^3 + 8^3.

Chai Wah Wu, Table of n, a(n) for n = 1..33

Cf. A003325, A050787, A068601.

isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
for(n=1, 1e7, if(isA003325(n) && isA003325(n^3-1), print1(n, ", ")));

a(8)-a(16) from Chai Wah Wu, Apr 17 2016

a(17)-a(25) from Chai Wah Wu, Jul 21 2025

A271876 Numbers n such that 3^n is not of the form x^3 + y^3 + z^3 where x, y, z > 0.

Original entry on oeis.org

0, 2, 3, 5, 8, 11, 14, 17, 20

Offset: 1

Altug Alkan, Apr 16 2016

This sequence is finite.
If 3^n = x^3 + y^3 + z^3, then 3^(n+3) = 3^3 * 3^n = 3^3 * (x^3 + y^3 + z^3) = (3*x)^3 + (3*y)^3 + (3*z)^3. So if we can find the integer n such that 3^n, 3^(n+1) and 3^(n+2) are sums of 3 positive cubes, we will exactly know that this sequence must be finite. Obviously, the minimum value of n such that 3^n, 3^(n+1) and 3^(n+2) are sums of 3 positive cubes determines the last term of this sequence. Since we can solve the 3^n = x^3 + y^3 + z^3 for n = 21, 22, 23 as in example section, all terms of this sequence is computed in limited range that is 0 <= n <= 20.
Corresponding 3^n values are 1, 9, 27, 243, 6561, 177147, 4782969, 129140163, 3486784401.

			21 is not a term because 3^21 = (3^5)^3 + (6*3^5)^3 + (8*3^5)^3.
22 is not a term because 3^22 = (3^7)^3 + (3^7)^3 + (3^7)^3.
23 is not a term because 3^23 = 1658^3 + 3202^3 + 3843^3.

Cf. A000244, A003072.

Select[Range[0, 20], Length[PowersRepresentations[3^#, 3, 3] /. {{0, } -> Nothing}] == 0 &] (* Michael De Vlieger, Apr 16 2016 *)

A273219 Taxi-cab numbers (A001235) that are the sum of three positive cubes.

Original entry on oeis.org

216027, 262656, 515375, 1092728, 1331064, 1533357, 1728216, 1845649, 2101248, 2515968, 2562112, 2622104, 2864288, 3511872, 3551112, 4033503, 4123000, 4511808, 4607064, 5004125, 5462424, 5832729, 6017193, 7091712, 7882245, 8491392, 8493039, 8494577, 8741824

Offset: 1

Altug Alkan, May 18 2016

nonn

A001235(158) = 10702783 = 7*13*337*349 is the least squarefree term of this sequence.

			216027 is a term because 216027 = 3^3 + 60^3 = 22^3 + 59^3 = 11^3 + 42^3 + 52^3.
262656 is a term because 262656 = 8^3 + 64^3 = 36^3 + 60^3 = 15^3 + 42^3 + 57^3.
515375 is a term because 515375 = 15^3 + 80^3 = 54^3 + 71^3 = 30^3 + 62^3 + 63^3.

Chai Wah Wu, Table of n, a(n) for n = 1..16254

Cf. A001235, A003072.

cp's OEIS Frontend

A069137 Numbers which are sums of neither 1, 2, 3, 4, 5 or 6 nonnegative cubes.

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A226955 Number of representations of n! as a sum of 3 positive cubes.

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A084355 Least number of positive cubes needed to represent n!.

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A085334 Numbers which are neither sums of 2,3,4,5 or that of 6 nonnegative cubes.

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A085336 Numbers which are sums of two and also sums of three positive cubes.

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Mathematica

A104054 Numbers which are the sum of three positive cubes and divisible by 31.

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Mathematica

A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.

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A271717 Integers k such that both k and k^3-1 are the sum of two positive cubes (see A003325).

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A271876 Numbers n such that 3^n is not of the form x^3 + y^3 + z^3 where x, y, z > 0.

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