A114923 -id:A114923 - cp's OEIS frontend

A023042 Numbers whose cube is the sum of three distinct nonnegative cubes.

6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 57, 58, 60, 63, 66, 67, 69, 70, 71, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 103, 105, 106, 108, 110, 111, 112, 113

Offset: 1

N. J. A. Sloane

nonn

Numbers w such that w^3 = x^3+y^3+z^3, x>y>z>=0, is soluble.
A226903(n) + 1 is an infinite subsequence parametrized by Shiraishi in 1826. - Jonathan Sondow, Jun 22 2013
Because of Fermat's Last Theorem, sequence lists numbers w such that w^3 = x^3+y^3+z^3, x>y>z>0, is soluble. In other words, z cannot be 0 because x and y are positive integers by definition of this sequence. - Altug Alkan, May 08 2016
This sequence is the same as numbers w such that w^3 = x^3+y^3+z^3, x>=y>=z>0, is soluble as Legendre showed that a^3+b^3=2*c^3 only has the trivial solutions a = b or a = -b (see Dickson's History of the Theory of Numbers, vol. II, p. 573). - Chai Wah Wu, May 13 2017

			20 belongs to the sequence as 20^3 = 7^3 + 14^3 + 17^3.

Ya. I. Perelman, Algebra can be fun, pp. 142-143.

Nathaniel Johnston and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (1..1500 from Nathaniel Johnston)
A. Russell and C. E. Gwyther, The Partition of Cubes, The Mathematical Gazette, 21 (1937), pp. 33-35.
Index to sequences related to Diophantine equations (3,1,3)

Cf. A001235, A003072, A114923, A225908, A226903.

for w from 1 to 113 do for z from 0 to w-1 do bk:=0: for y from z+1 to w-1 do for x from y+((w+z) mod 2) to w-1 by 2 do if(x^3+y^3+z^3=w^3)then printf("%d, ",w); bk:=1: break: fi: od: if(bk=1)then break: fi: od: if(bk=1)then break: fi: od: od: # Nathaniel Johnston, Jun 22 2013

Select[Range[200], n |-> Length[PowersRepresentations[n^3, 3, 3]] > 1] (* Paul C Abbott, May 07 2025 *)

has(n)=my(L=sqrtnint(n-1,3)+1, U=sqrtnint(4*n,3)); fordiv(n,m, if(L<=m && m<=U, my(ell=(m^2-n/m)/3); if(denominator(ell)==1 && issquare(m^2-4*ell), return(1)))); 0
list(lim)=my(v=List(),a3,t); lim\=1; for(a=2,sqrtint(lim\3), a3=a^3; for(b=if(a3>lim, sqrtnint(a3-lim-1,3)+1,1), a-1, t=a3-b^3; if(has(t), listput(v,a)))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

A258865 Numbers that are a sum of the cubes of three primes.

Original entry on oeis.org

24, 43, 62, 81, 141, 160, 179, 258, 277, 359, 375, 378, 397, 476, 495, 593, 694, 713, 811, 1029, 1347, 1366, 1385, 1464, 1483, 1581, 1682, 1701, 1799, 2017, 2213, 2232, 2251, 2330, 2349, 2447, 2548, 2567, 2665, 2670, 2689, 2787, 2883, 3005, 3536, 3555

Offset: 1

R. J. Mathar, Jun 12 2015

nonn

The subsequence of cubes in the sequence starts 505^3, 535^3, 709^3, 865^3, 1033^3, 1037^3, 1067^3, 1133^3, 1513^3, ... See A258262.

			2^3+2^3+2^3=24. 2^3+2^3+3^3=43. 2^3+3^3+3^3=62. 3^3+3^3+3^3=81.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A138854, A114923.

Cf. A030078, A258262 (subsequence).

import Data.Set (singleton, deleteFindMin, fromList)
import qualified Data.Set as Set (union)
import qualified Data.List.Ordered as List (union)
a258865 n = a258865_list !! (n-1)
a258865_list = tail $ f (singleton 1) 1 [] [] a030078_list where
   f s z vs qcs pcs'@(pc:pcs)
     | m < z = m : f s' z vs qcs pcs'
     | otherwise = f (Set.union s $ fromList $ map (+ pc) ws)
                     pc ws (pc:qcs) pcs
     where ws = List.union vs $ map (+ pc) (pc : qcs)
           (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 13 2015

A258865 := proc(lim)
    local a,p,q,r ;
    a := {} ;
    p := 2 ;
    while p^3 < lim do
        q := p ;
        while p^3 +q^3< lim do
            r := q ;
            while p^3+q^3+r^3 <= lim do
                a := a union {p^3+q^3+r^3} ;
                r := nextprime(r) ;
            end do:
            q := nextprime(q) ;
        end do:
        p := nextprime(p) ;
    end do ;
    a ;
end proc:
A258865(30000) ;

lim = 15; Take[Sort@ DeleteDuplicates[Total /@ (Tuples[Prime@ Range@ lim, 3]^3)], 3 lim] (* Michael De Vlieger, Jun 12 2015 *)

list(lim)=my(v=List(), P=apply(p->p^3,primes(sqrtnint(lim\=1,3)))); foreach(P,p, foreach(P,q, my(s=p+q,t); for(i=1,#P, t=s+P[i]; if(t>lim,break); listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 09 2021

a(n) = A030078(i)+A030078(j)+A030078(k) for some triple (i,j,k).

By a counting argument a(n) >> n log^3 n and hence the sequence is of density 0. - Charles R Greathouse IV, Aug 09 2021

A113490 Semiprimes a such that there exist three semiprimes b, c and d with a^3=b^3+c^3+d^3.

Original entry on oeis.org

206, 519, 703, 869, 1418, 1923, 1945, 2066, 2095, 2127, 2446, 2759, 2867, 2881, 2901, 2913, 2974, 3099, 3155, 3207, 3383, 3398, 3545, 3649, 3777, 3814, 3898, 4435, 4766, 4778, 4873, 4963, 5091, 5105, 5165, 5534, 5582, 5638, 5771, 5834, 5855, 6033, 6098

Offset: 1

Jonathan Vos Post, Jan 09 2006

nonn

This is the semiprime analog of A114923.

There are only two such semiprimes < 10^4 with more than one solution: 2095 and 9897.

			206^3 = 35^3 + 77^3 + 202^3.
519^3 = 4^3 + 303^3 + 482^3
703^3 = 111^3 + 291^3 + 685^3.
869^3 = 466^3 + 629^3 + 674^3.
2095^3 = 339^3 + 753^3 + 2059^3 = 543^3 + 1119^3 + 1969^3 (two ways).
9897^3 = 537^3 + 1454^3 + 9886^3 = 2071^3 + 3183^3 + 9755^3 (two ways).
Each of these numbers (other than the exponent 3) is a semiprime (A001358).

Cf. A001358, A114923.

Extended by Ray Chandler, Jan 20 2006

A384553 Primes p for which there exists more than one triple of primes q, r, s such that p^3 = q^3 + r^3 + s^3.

Original entry on oeis.org

28477, 33199, 49069, 234181, 300239, 403549, 463501, 958933, 982337, 1044227, 1352873, 1385861, 1713121, 1834321, 1994911, 2364673, 2531687, 2839927, 3048691, 3364553, 3546031, 3640543, 3897739, 3941711, 4000907, 4264219, 4273459, 4594399, 4599709, 4620037, 4924979

Offset: 1

Zhining Yang, Jun 03 2025

nonn

			28477^3 = 3739^3 + 17203^3 + 26183^3 = 10781^3 + 11071^3 + 27361^3.
33199^3 = 2833^3 + 19081^3 + 30941^3 = 15187^3 + 24197^3 + 26647^3.
49069^3 =  661^3 + 37441^3 + 40343^3 = 22307^3 + 37243^3 + 38119^3.

Zhining Yang, Table of n, a(n) for n = 1..52

Cf. A008917, A114923.

cp's OEIS Frontend

A023042 Numbers whose cube is the sum of three distinct nonnegative cubes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A258865 Numbers that are a sum of the cubes of three primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A113490 Semiprimes a such that there exist three semiprimes b, c and d with a^3=b^3+c^3+d^3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A384553 Primes p for which there exists more than one triple of primes q, r, s such that p^3 = q^3 + r^3 + s^3.

Views

Author

Keywords

Examples

Links

Crossrefs