A022552 -id:A022552 - cp's OEIS frontend

A275168 Positive integers not of the form x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

6, 18, 23, 41, 42, 59, 78, 86, 96, 114, 115, 123, 142, 187, 195, 205, 213, 214, 240, 261, 262, 266, 303, 322, 329, 330, 383, 423, 478, 501, 510, 581, 610, 618, 642, 682, 690, 698, 761, 774, 807, 865, 870, 906, 959, 963, 990, 1206, 1222, 1230, 1234, 1302, 1312, 1314, 1320, 1346, 1411, 1697, 1706, 1781

Offset: 1

Zhi-Wei Sun, Jul 18 2016

nonn

Conjecture: The sequence has totally 150 terms as listed in the b-file the largest of which is 182842. Thus any integer n > 182842 can be written as x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.
We note that the sequence has no term greater than 182842 and not exceeding 10^6.
See also A275169 for a similar conjecture.
It is known that for any positive integers a,b,c there are infinitely many positive integers not of the form a*x^2 + b*y^2 + c*z^2 with x,y,z nonnegative integers.

			a(1) = 6 since 1 = 0^3 + 3*0^2 + 1^2, 2 = 1^3 + 3*0^2 + 1^2, 3 = 0^3 + 3*1^2 + 0^2, 4 = 0^3 + 3*1^2 + 1^2, 5 = 1^3 + 3*1^2 + 1^2, but 6 cannot be written as x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275169.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[m-x^3-3*y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/3]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,1800}]

A299796 Largest number k that is not the sum of 3 squares and a nonnegative (2n+5)-th power.

Original entry on oeis.org

2160, 19568, 177136, 1594304, 14348784, 129140160, 1162261440, 10460353136, 94143178736, 847288609392, 7625597484912, 68630377364848, 617673396283888, 5559060566555504, 50031545098999664, 450283905890997360, 4052555153018976240, 36472996377170786288

Offset: 1

XU Pingya, Feb 19 2018

nonn

Subsequence of A004215.

			2160 is not the sum of 3 squares and a nonnegative 7th power, and all integers larger than 2160 can be represented as the sum of 3 squares and a nonnegative 7th power (see A297970). Thus a(1) = 2160.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A004215, A022552, A297970.

a[n_]:= Max@Table[4^k (8 Floor[3^(2 n + 5) / 2^(2 k + 3) - 7 / 8] + 7),  {k, 2, n + 1}] Table[a[n], {n, 18}] (* XU Pingya, Nov 30 2018 *)

a(n) = max_{2<=i<=n+1, j>=0} {4^i*(8j+7) : 4^i*(8j+7) < 3^(2n+5)}.

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]

A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

Original entry on oeis.org

1, 2, 3, 2, 2, 2, 2, 1, 2, 4, 5, 3, 2, 3, 2, 1, 3, 5, 6, 3, 3, 3, 2, 0, 2, 5, 6, 5, 4, 5, 2, 2, 4, 5, 6, 4, 6, 6, 4, 2, 4, 6, 4, 4, 4, 7, 3, 2, 4, 3, 5, 4, 7, 8, 5, 3, 3, 3, 5, 5, 5, 6, 4, 3, 6, 7, 8, 7, 5, 7, 4, 2, 7, 9, 10, 4, 5, 7, 3, 3, 9, 10, 8, 5, 4, 7

Offset: 0

XU Pingya, Jan 08 2018

nonn

For n <= 6 * 10^7, except for a(23) = 0, all a(n) > 0.

First occurrence of k beginning with 0: 23, 7, 1, 2, 9, 10, 18, 45, 53, 73, 74, 101, 125, 146, 165, 197, ..., . - Robert G. Wilson v, Jan 14 2018

			2 = 0^2 + 0^2 + 1^3 + 1^3 = 0^2 + 1^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 0^3, a(2) = 3.
10 = 0^2 + 1^2 + 1^3 + 2^3 = 0^2 + 3^2 + 0^3 + 1^3 = 1^2 + 1^2 + 0^3 + 2^3 = 1^2 + 3^2 + 0^3 + 0^3 = 2^2 + 2^2 + 1^3 + 1^3, a(10) = 5.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A000164, A002635, A022552, A274274.

a[n_] := Sum[If[x^2 + y^2 + z^3 + u^3 == n, 1, 0], {x, 0, n^(1/2)}, {y, x, (n - x^2)^(1/2)}, {z, 0, (n - x^2 - y^2)^(1/3)}, {u, z, (n - x^2 - y^2 - z^3)^(1/3)}]; Table[a[n], {n, 0, 86}]

A319052 Numbers that are not the sum of {2 squares, a nonnegative cube, and a nonnegative k-th power with k >= 17}.

Original entry on oeis.org

23, 71, 120, 312, 455, 2136, 99295

Offset: 1

XU Pingya, Sep 08 2018

Subsequence of A022552.
Except for the above 7 terms, the remaining 427 numbers in A022552 can be expressed as the sum of two squares, a nonnegative cube and a nonnegative k-th power. So a(n) has only 7 terms, until n = 10^10.
Also, for n <= 6*10^7, when k = 3, the number of such forms is only 23; when 4 <= k <= 5, only 23 and 71; when k = 6, only 23, 71 and 455; when 7 <= k <= 8, only 23, 71 and 120; when 9 <= k <= 11, only 23, 71, 120, 312 and 455; when 12 <= k <= 16, only 23, 71, 120, 312, 455 and 2136.

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

Cf. A022552, A297930.

n=100000;
t=Union@Flatten@Table[x^2+y^2+z^3+w^17, {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/3)}, {w,0,(n-x^2-y^2-z^3)^(1/17)}];
Complement[Range[0,n], t]

A330708 Numbers that are not the sum of 2 nonzero squares and a positive cube.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 12, 15, 17, 20, 22, 23, 24, 31, 36, 39, 43, 50, 55, 57, 63, 65, 70, 71, 78, 87, 94, 103, 111, 113, 115, 119, 120, 134, 139, 141, 148, 160, 167, 169, 185, 204, 211, 254, 263, 267, 279, 283, 286, 302, 311, 312, 331, 335, 342, 349, 379, 391

Offset: 1

XU Pingya, Jun 08 2020

nonn

A022552 is a subsequence.

a(490) = A022552(434) = 5042631. No more terms <= 4 * 10^7.

Robert Israel, Table of n, a(n) for n = 1..490

Cf. A022552, A000534, A004214.

N:= 500: # for terms <= N
G1:= add(x^(i^2), i=1..floor(sqrt(N))):
G2:= add(x^(i^3), i=1..floor(N^(1/3))):
G:= expand(G1^2*G2):
select(t -> coeff(G,x,t)=0, [$0..N]); # Robert Israel, Jun 12 2020

m = 0;
n = 400.;
t = Union@Flatten@Table[x^2 + y^2 + z^3, {x, (n/2)^(1/2)}, {y, x, (n - x^2)^(1/2)}, {z, If[x^2 + y^2 < m, Floor[(m - 1 - x^2 - y^2)^(1/3)] + 1, 1], (n - x^2 - y^2)^(1/3)}];
Complement[Range[m, n], t]

A214922 Numbers of the form x^2 + y^2 + z^3 + w^3 (x, y, z, w >= 0).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54

Offset: 1

Philippe Deléham, Jul 20 2012

nonn

Conjecture: 23 is the only number not in this sequence.
Not the same as A004830: 239 is a term of this sequence but not of A004830. - R. J. Mathar and Joerg Arndt, Jul 28 2012
There are no other missing numbers from 24 to 10^8. - Giovanni Resta, Oct 12 2019

			22 = 2^2 + 4^2 + 1^3 + 1^3, 22 is in this sequence.

Cf. A001481, A004999, A022549, A022550, A022551, A022552.

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

A305553 Numbers that are not the sum of 2 squares and a 4th power.

Original entry on oeis.org

7, 12, 15, 22, 23, 28, 31, 39, 43, 44, 47, 55, 60, 63, 67, 70, 71, 76, 78, 79, 87, 92, 93, 95, 103, 108, 111, 112, 119, 124, 127, 135, 140, 143, 151, 156, 159, 167, 168, 172, 175, 177, 183, 184, 188, 191, 192, 199, 204, 207, 214, 215, 220, 223, 231, 236

Offset: 1

XU Pingya, Jun 20 2018

nonn

Numbers of the form 4*A017101(k) are terms of this sequence.

m is a term iff 16m is also.

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

Subsequence of A000037, A140823 and A022544.
A004215 and A214891 are subsequences.
Cf. A017101, A022552.

n=239;
t=Union@Flatten@Table[x^2+y^2+z^4, {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/4)}];
Complement[Range[0,n], t]

cp's OEIS Frontend

A275168 Positive integers not of the form x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

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A299796 Largest number k that is not the sum of 3 squares and a nonnegative (2n+5)-th power.

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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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A297930 Number of partitions of n into 2 squares and 2 nonnegative cubes.

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A319052 Numbers that are not the sum of {2 squares, a nonnegative cube, and a nonnegative k-th power with k >= 17}.

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A330708 Numbers that are not the sum of 2 nonzero squares and a positive cube.

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A214922 Numbers of the form x^2 + y^2 + z^3 + w^3 (x, y, z, w >= 0).

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

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