A022566 -id:A022566 - cp's OEIS frontend

A004826 Numbers that are the sum of at most 4 positive cubes.

0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 17, 18, 24, 25, 27, 28, 29, 30, 32, 35, 36, 37, 43, 44, 51, 54, 55, 56, 62, 63, 64, 65, 66, 67, 70, 72, 73, 74, 80, 81, 82, 88, 89, 91, 92, 93, 99, 100, 107, 108, 118, 119, 125, 126, 127, 128, 129, 130, 133, 134, 135

Offset: 1

N. J. A. Sloane

nonn

Stated in Lee, p. 1: It is now known that when N is sufficiently large, the number of positive integers at most N that fail to be written in such a way (A022566) is slightly smaller than N^(37/42). Since any integer congruent to 4 (mod 9) is never a sum of three cubes, the number of summands here cannot in general be reduced. But of those four cubes, two of which (minicubes) need be at most N^theta, as long as theta >= 192/869. An asymptotic formula for the number of such representations is established when 1/4 < theta < 1/3. - Jonathan Vos Post, Jun 29 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Siu-lun Alan Lee, On Waring's Problem: Two Cubes and Two Minicubes, arXiv:1006.5142 [math.NT], 2010.
G. Villemin's Almanach of Numbers, Sum of Four Cubes (0 through 100).
Index entries for sequences related to sums of cubes

Cf. A022566 (Numbers that are not the sum of 4 nonnegative cubes). - Jonathan Vos Post, Jun 29 2010

Reap[For[k = 0, k <= 200, k++, If[PowersRepresentations[k, 4, 3] != {}, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Oct 05 2018 *)

A022555 Positive integers that are not the sum of two nonnegative cubes.

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

N. J. A. Sloane

nonn

Omits the positive cubes (A000578) since m^3 = 0^3 + m^3, so is different from A057903.

Complement of A004999. Cf. A004825, A022561, A022566, A057903, A000578.

read transforms; cub := series(add(q^(n^3),n=0..100),q,1000001); t1 := series(cub^2,q,2000); t2 := POWERS(t1,q,2000); COMPl(t2);

r[n_] := Reduce[x >= 0 && y >= 0 && n == x^3 + y^3, {x, y}, Integers]; Select[ Range[80], r[#] === False &]  (* Jean-François Alcover, Nov 06 2012 *)
Select[Range@100,PowersRepresentations[#,2,3]=={}&] (* A much faster solution given by Giovanni Resta, Nov 06 2012 *)

Edited by N. J. A. Sloane, Sep 28 2007

A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

Original entry on oeis.org

112, 240, 368, 496, 624, 752, 880, 1008, 1136, 1264, 1392, 1520, 1648, 1776, 1904, 2032, 2160

Offset: 1

XU Pingya, Jan 10 2018

The last term in this sequence is 2160. The reasons are as follows (let b, c, d, i, j, k, m, r, s, t, w, x, y and z be nonnegative integers).
For the Diophantine equation x^2 + y^2 + z^2 + w^7 = m:
(1) If m is not of the form 4^c * (8b + 7), then it follows from Legendre's three-square theorem that the equation has a solution with w = 0.
(2) 8b + 7 - 1^7 = 8b + 6. Then m = 8b + 7, the equation has a solution with w = 1.
(3) 4 * (8b + 7) - 1^7 = (8 * (4b + 3) + 3) = 8d + 3. Then m = 4 * (8b + 7), the equation has a solution with w = 1.
(4) For b >= 17, 16 * (8b + 7) - 3^7 = 8 * (16 * (b - 17) + 12) + 5 = 8i + 5. Then m = 16 * (8b + 7) and b >= 17, the equation has a solution with w = 3.
(5) 4^3 * (8b + 7) - 2^7 = 4^3 * (8b + 5). Then m = 4^3 * (8b + 7), the equation has a solution with w = 2. And 4^3 * (8b + 7) - 3^7 = 8 * (4^3 * (b - 4) + 38) + 5 = 8j + 5. Then m = 4^3 * (8b + 7) and b >= 4, the equation has a solution with w = 3.
(6) 4^4 * (8b + 7) - 2^7 = 4^3 * (8 * (4b + 3) + 3) = 4^3 * (8k + 3). 4^4 * (8b + 7) - 3^7 = 8 * (256b - 217) + 3 = 8r + 3. Then m = 4^4 * (8b + 7), the equation has a solution with w = 2 and when b > 0, the equation has a solution with w = 3.
(7) When c >= 5, 4^c * (8b + 7) - 2^7 = 4^3 * (8 * (b * 4^(c - 3) + 14 * 4^(c - 5) + 5) = 4^3 * (8s + 5). 4^c * (8b + 7) - 3^7 = 8 * (b * 4^(c - 3) + 14 * 4^(c - 3) - 273) + 3 = 8t + 3. Then n = 4^c * (8b + 7), the equation has solutions with w = 2 and 3.
In short, except for the 17 numbers in the sequence, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative 7th power.

Wikipedia, Legendre's three-square theorem

Finite subsequence of A004215 and A296185.

Cf. A004771, A022552, A022557, A022561, A022566, A111151.

t1={};
Do[Do[If[x^2+y^2+z^2+w^7==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/7)}], {n,0,3000}];
t2={};
Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,3000}];
t2

a(n) = 128n - 16 = 16 * A004771(n - 1), 1 <= n <= 17.

A267826 Numbers not of the form w^3 + 2x^3 + 3y^3 + 4*z^3, where w, x, y and z are nonnegative integers.

Original entry on oeis.org

18, 22, 39, 60, 63, 74, 76, 77, 100, 103, 106, 107, 117, 126, 178, 180, 201, 215, 228, 230, 245, 271, 289, 291, 295, 315, 341, 356, 357, 393, 413, 419, 420, 480, 481, 523, 559, 606, 616, 671, 673, 705, 854, 855, 963, 980, 981, 998, 1103, 1121, 1130, 1298, 1484, 1510, 1643, 1729, 1849, 1916, 1934, 1946

Offset: 1

Zhi-Wei Sun, Apr 07 2016

nonn

Conjecture: The sequence has exactly 122 terms the last of   which is a(122) = 41405.
We have verified that there are no terms between 41406 and 2*10^5.
The conjecture implies that {P(v)+w^3+2*x^3+3*y^3+4*z^3: w,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(v) is among the polynomials a*v^3 (a = 1,5,6,7,9,10,12,15,18), b*v^4 (b = 1,2,3,5,6,12,18), c*v^5 (c = 1,2,5,12) and d*v^k (d = 5,12; k = 6,7). Moreover, it also implies that {8*t+w^3+2*x^3+3*y^3+4*z^3: t = 0,1; w,x,y,z = 0,1,2,...} = {0,1,2,...}. If a,b,c,d and m are positive integers with {m*t+a*w^3+b*x^3+c*y^3+d*z^3: t = 0,1; w,x,y,z = 0,1,2,...} = {0,1,2,...}, then we must have m = 8 and {a,b,c,d} = {1,2,3,4}.

			a(1) = 18 since it is the first nonnegative integer not in the set {w^3 + 2*x^3 + 3*y^3 + 4*z^3: w,x,y,z = 0,1,2,...}.

Zhi-Wei Sun, Table of n, a(n) for n = 1..122
Zhi-Wei Sun, Universal sums u^3+a*v^3+b*x^3+c*y^3+d*z^3 with u, v, x, y, z nonnegative integers, a message to Number Theory Mailing List, April 3, 2016.

Cf. A000578, A002804, A022566, A267861, A271099, A271169, A271237.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
n=0;Do[Do[If[CQ[m-4*z^3-3y^3-2x^3],Goto[aa]],{z,0,(m/4)^(1/3)},{y,0,((m-4z^3)/3)^(1/3)},{x,0,((m-4z^3-3y^3)/2)^(1/3)}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,0,1946}]

A296185 Numbers that are not the sum of 3 squares and an 8th power.

Original entry on oeis.org

112, 240, 368, 496, 624, 752, 880, 1008, 1136, 1264, 1392, 1520, 1648, 1776, 1904, 2032, 2160, 2288, 2416, 2544, 2672, 2800, 2928, 3056, 3184, 3312, 3440, 3568, 3696, 3824, 3952, 4080, 4208, 4336, 4464, 4592, 4720, 4848, 4976, 5104, 5232, 5360, 5488, 5616

Offset: 1

XU Pingya, Jan 13 2018

nonn

When m is in this sequence, 9m and m^9 are also in this sequence.
For nonnegative integers a, b, k, n, x, y, z and w, n = x^2 + y^2 + z^2 + w^8 if and only if n is not of the form 4^(4k + 2) * (8b + 7).
1. If n is not of the form 4^a * (8b + 7), then it follows from Legendre's three-square theorem that the equation x^2 + y^2 + z^2 + w^8 = n has a solution with w = 0.
2. If n = 4^a * (8b + 7), then (c, d and j are nonnegative integers):
(1) If a = 4k, then n - (2^k)^8 = 4^(4k) * (8b + 6), and the equation has a solution with w = 2^k.
(2) If a = 4k + 1, then n - (2^k)^8 = 4^(4k) * (32b + 27) is of the form 4^c * (8d + 3), and the equation has a solution with w = 2^k.
(3) If a = 4k + 3, then n - (2^(k + 1))^8 = 4^a * (8b + 3), and the equation has a solution with w = 2^(k + 1).
(4) If a = 4k + 2 and w = 2j + 1, then n == 0 (mod 8), w^8 == 1 (mod 8), and n - w^8 is number of the form 8c + 7. I.e., the equation does not have a solution with w odd.
If a = 4k + 2 and w = 4j + 2, then n - w^8 = 16 * (4^(4k) * (8b + 7) - 16 * (2j + 1)^8) = 4^4 * (4^(4k - 4) * (8b + 7) - (2j + 1)^8). When k = 0, n - w^8 is a number of the form 16 * (8c + 7); when k > 0, n - w^8 is a number of the form 4^4 * (8d + 7). Therefore, the equation does not have a solution with w = 4j + 2. Similarly, it can be proved that there is no solution with w = 4j.

Wikipedia, Legendre's three-square theorem

Cf. A004215, A022552, A022557, A022561, A022566, A111151.

t1={};
Do[Do[If[x^2+y^2+z^2+w^8==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/8)}], {n,0,5700}];
t2={};
Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,5700}];
t2

from itertools import count, islice
def A296185_gen(): # generator of terms
    for k in count(0):
        r = 1<<((k<<1)+1<<2)
        yield from range(7*r,r*((r<<8)+7),r<<3)
A296185_list = list(islice(A296185_gen(),44)) # Chai Wah Wu, May 21 2025

A296579 Numbers that are not the sum of 3 squares and a nonnegative 9th power.

Original entry on oeis.org

112, 240, 368, 448, 496, 624, 752, 880, 960, 1008, 1136, 1264, 1392, 1472, 1520, 1648, 1776, 1904, 1984, 2032, 2160, 2288, 2416, 2496, 2544, 2672, 2800, 2928, 3008, 3056, 3184, 3312, 3440, 3520, 3568, 3696, 3824, 3952, 4032, 4080, 4208, 4336, 4464, 4544, 4592

Offset: 1

XU Pingya, Jan 30 2018

a(n) consists of the number of forms 16*(8i + 7) (0 <= i <= 152) and 64*(8j + 7) (0 <= j <= 37).

The last term in this sequence is a(191) = 19568 = 16*(8*152 + 7) (see A297970).

Wikipedia, Legendre's three-square theorem

Finite subsequence of A004215.
A297970 is a subsequence.
Cf. A004771, A022552, A022557, A022561, A022566, A111151.

t1=Table[4^2*(8j+7), {j,0,152}];
t2=Table[4^3*(8j+7), {j,0,37}];
t=Union[t1, t2]

A297931 Numbers that are not the sum of a square and 3 nonnegative cubes.

Original entry on oeis.org

15, 22, 23, 48, 86, 94, 112, 120, 139, 184, 203, 211, 230, 237, 263, 301, 309, 312, 335, 373, 399, 1014, 1056, 1455, 1644, 2029, 2272, 2658, 3309, 3469, 4019, 6502, 11101

Offset: 1

XU Pingya, Jan 08 2018

nonn

After 11101, there are no more terms up to 570000.

No more terms < 10^10; is this sequence finite? - Mauro Fiorentini, Jan 26 2019

Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A004215, A022552, A022557, A022561, A022566, A111151.

t1={};
Do[Do[If[x^2+y^2+z^2+w^3==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/3)}], {n,0,5.7*10^5}];
t2={};
Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,5.7*10^5}];
t2

A332664 a(n) = number of nonnegative integers that are not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.

Original entry on oeis.org

0, 2, 14, 115, 116, 109, 245, 381, 1387, 913, 1234, 1552, 2103, 2838, 3036, 3384, 4693, 5405, 8304, 9088, 11089, 13289, 15815, 18619, 20979, 22755, 24107, 24984, 25548

Offset: 2

XU Pingya, Feb 18 2020

a(2) = 0 by a theorem of Zhi-Wei Sun, see A273915. All terms beyond a(2) are conjectures and have only been checked to 4*10^9.

			a(2) = 0, since any nonnegative integer k is the sum of 3 squares and a nonnegative 5th power (see A273915).
a(4) = 14. Since any nonnegative integer k (<= 4*10^9) is the sum of {2 squares, a nonnegative 5th power, and a 4th power}, except for 14 numbers: 23, 44, 71, 79, 215, 383, 863, 1439, 1583, 1727, 1759, 1919, 2159, 2543.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.

Cf. A022566, A111151, A273915, A319052, A322122.

a(5)
Do[m=1000000 (k-1)+1; n=1000000 k;
  t=Union@Flatten@Table[x^2 + y^2 + z^5 + w^5,
{x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/5)},
{w, If[x^2 + y^2 + z^5 < m, Floor[(m-1-x^2-y^2-z^5)^(1/5)] + 1, z], (n-x^2-y^2-z^5)^(1/5)}];
  b=Complement[Range[m, n], t];
  Print[Length@b], {k,4000}]

A332665 a(n) = largest number k that is not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.

Original entry on oeis.org

71, 2543, 146687, 55871, 63703, 353087, 606239, 21073228, 4606428, 19212172, 16301375, 88507864, 228238087, 126691212, 121909855, 366845415, 448525727, 1818767247, 1319413228, 341968924, 673747279, 188739607, 2366756975, 1770393855, 411629407, 1240489516

Offset: 3

XU Pingya, Feb 19 2020

nonn

All terms are conjectural and have only been checked up to 4*10^9.

			a(4) = 2543, since 2543 is the greatest number such that is not the sum of {2 squares, a nonnegative 5th power, and a 4th power} (at least up to 4*10^9).

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.

Cf. A022566, A111151, A299796.

cp's OEIS Frontend

A004826 Numbers that are the sum of at most 4 positive cubes.

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A022555 Positive integers that are not the sum of two nonnegative cubes.

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A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

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A267826 Numbers not of the form w^3 + 2*x^3 + 3*y^3 + 4*z^3, where w, x, y and z are nonnegative integers.

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A296185 Numbers that are not the sum of 3 squares and an 8th power.

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A296579 Numbers that are not the sum of 3 squares and a nonnegative 9th power.

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A297931 Numbers that are not the sum of a square and 3 nonnegative cubes.

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A332664 a(n) = number of nonnegative integers that are not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.

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A332665 a(n) = largest number k that is not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.

A267826 Numbers not of the form w^3 + 2x^3 + 3y^3 + 4*z^3, where w, x, y and z are nonnegative integers.