A003327 -id:A003327 - cp's OEIS frontend

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

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

A274334 Numbers whose cube is the sum of 4 positive cubes.

Original entry on oeis.org

7, 12, 13, 14, 18, 20, 21, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93

Offset: 1

Altug Alkan, Jun 23 2016

Conjecture: a(n) = n + 21 for all n > 50. - Charles R Greathouse IV, Jan 14 2017

			7 is a term because 7^3 = 1^3 + 1^3 + 5^3 + 6^3.

Index to sequences related to diophantine equations (3,1,4)

Cf. A003327.

Name edited by Michel Marcus, Jul 30 2025

A085320 First difference sequence of A003349, i.e., consecutive differences between those consecutive numbers which are sums of four 5th powers.

Original entry on oeis.org

31, 31, 31, 31, 118, 31, 31, 31, 149, 31, 31, 180, 31, 211, 55, 31, 31, 31, 149, 31, 31, 180, 31, 211, 297, 31, 31, 180, 31, 211, 539, 31, 24, 31, 31, 31, 94, 55, 31, 31, 180, 31, 211, 242, 55, 31, 31, 180, 31, 211, 539, 31, 211, 781, 55, 31, 31, 180, 31, 211, 539, 31

Offset: 1

Labos Elemer, Jul 01 2003

nonn

			35 = 1 + 1 + 1 + 32, 66 = 32 + 32 + 1 + 1, a(1) = 66 - 33 = 31.
Certain differences occur rather consequently like 31, 55, 113, 180, 207, 211, 539, etc.;
Distance of closest observed neighbors equals 2 like those of 33858 and 33856.

Cf. A003349, A003327-A003399.

{m=12, k=5, m^k}; t=Union[Flatten[Table[Table[Table[Table[w^k+q^k+t^k+u^k, {w, 1, m}], {q, 1, m}], {t, 1, m}], {u, 1, m}]]]; Length[t]; dt=Delete[ -RotateRight[t]+t, 1]; Sort[dt]

A085337 Numbers which are sums of two, three and four cubes.

Original entry on oeis.org

1072, 3402, 5256, 6244, 6867, 6984, 8576, 9288, 9728, 10261, 10656, 10745, 10773, 10989, 13392, 14167, 14364, 15093, 16480, 17603, 17920, 18305, 18369, 18648, 20026, 20320, 20538, 22016, 23085, 23408, 23625, 24416, 27133, 27216, 27792

Offset: 1

Labos Elemer, Jul 07 2003

nonn

			1072=729+343=1000+64+8=512+343+216+1

Cf. A003072, A003325, A003327.

Intersection of A003072, A003325 and A003327 sets.

More terms from Ray Chandler, Jul 21 2003

A085338 Numbers which are sums of two, three, four and also sums of five cubes.

Original entry on oeis.org

1072, 3402, 5256, 6867, 6984, 8576, 9288, 9728, 10261, 10656, 10745, 10773, 10989, 13392, 14167, 14364, 15093, 16480, 17603, 17920, 18305, 18369, 18648, 20026, 20320, 20538, 22016, 23085, 23408, 23625, 24416, 27133, 27216, 27792, 28000

Offset: 1

Labos Elemer, Jul 07 2003

nonn

			1072=729+343=1000+64+8=512+343+216+1=729+125+216+1+1

Intersection of A003072, A003325, A003327 and A003328 sets.

More terms from Ray Chandler, Jul 21 2003

A158794 Multiples of 4 which are not the sum of seven nonnegative cubes.

Original entry on oeis.org

212, 364, 420, 428

Offset: 1

Jonathan Vos Post, Mar 26 2009

Boklan and Elkies: It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.

Elkies [2010]: It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is congruent to 2 mod 4. This result, together with a recent proof for 4|N, shows that the conjecture is true for all even N. - Jonathan Vos Post, Sep 22 2010

U. V. Linnik: On the representation of large numbers as sums of seven cubes. Rec. Math. [=Mat. Sbornik] N.S. 12(54) (1943), 218-224.
L. E. Dickson: All integers except 23 and 239 are the sums of 8 cubes. Bull. Amer. Math. Soc. 45 (1939), 588-591.

E. Waring: Meditationes Algebraicae [3rd ed. (1782)]: an English translation of the work of Edward Waring, edited and translated from the Latin by Dennis Weeks. Providence, RI: Amer. Math. Soc., 1991. [MR1146921]
Kent D. Boklan and Noam D. Elkies, Every multiple of 4 except 212, 364, 420, and 428 is the sum of seven cubes, arXiv:0903.4503, Mar 26, 2009.
Noam D. Elkies, Every even number greater than 454 is the sum of seven cubes, Sep 21, 2010. [From _Jonathan Vos Post_, Sep 22 2010]

Cf. A000578, A003072, A003325, A003327, A003328, A003329, A057907, A122730.

Subsequence of A018888. - Charles R Greathouse IV, Apr 16 2010

Definition corrected by Jonathan Sondow, Mar 14 2014

A384132 Integers k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has no integer solutions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 21, 22, 25, 27, 29, 47, 58, 61, 71, 113, 121

Offset: 1

Zhining Yang, May 20 2025

Conjecture: a(27)=121 is the largest integer whose cube cannot be described as the sum of four distinct positive cubes.

			13 is not a term because 13^3 = 5^3 + 7^3 + 9^3 + 10^3 = 1^3 + 5^3 + 7^3 + 12^3.

Cf. A003327, A383877.

a=Select[Range@125,Length@Select[PowersRepresentations[#^3,4,3],0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]==0&]

cp's OEIS Frontend

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

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A274334 Numbers whose cube is the sum of 4 positive cubes.

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A085320 First difference sequence of A003349, i.e., consecutive differences between those consecutive numbers which are sums of four 5th powers.

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Mathematica

A085337 Numbers which are sums of two, three and four cubes.

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A085338 Numbers which are sums of two, three, four and also sums of five cubes.

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A158794 Multiples of 4 which are not the sum of seven nonnegative cubes.

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A384132 Integers k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has no integer solutions.

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