A111151 -id:A111151 - cp's OEIS frontend

A135911 Number of 4-tuples (x,y,z,t) of nonnegative integers such that x^2+y^3+z^4+t^5 = n.

1, 4, 6, 4, 2, 3, 3, 1, 1, 4, 6, 4, 2, 2, 1, 0, 2, 7, 8, 3, 1, 2, 1, 0, 2, 6, 7, 5, 5, 4, 1, 1, 4, 8, 7, 2, 3, 7, 5, 1, 2, 5, 5, 4, 6, 5, 1, 1, 3, 7, 7, 3, 5, 6, 2, 0, 2, 5, 5, 4, 5, 2, 0, 2, 6, 11, 9, 3, 4, 6, 2, 0, 2, 5, 5, 3, 4, 3, 1, 2, 4, 10, 11, 7, 6, 4, 2, 1, 1, 6, 9, 7, 5, 3, 1, 1, 4, 8, 8, 3, 4

Offset: 0

N. J. A. Sloane, Mar 07 2008

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
K. B. Ford, The representation of numbers as sums of unlike powers II, J. Amer. Math. Soc., 9 (1996), 919-940.

Cf. A045634, A135910, A135912.

Cf. A111151 (n such that a(n)=0).

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.

A351064 Minimal number of positive perfect powers, with different exponents, whose sum is n (considering only minimal possible exponents for bases equal to 1).

Original entry on oeis.org

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

Offset: 1

Alberto Zanoni, Feb 22 2022

nonn

Conjecture: the only numbers for which 5 addends are needed are 15, 23, 55, 62, 71.

The numbers mentioned in the conjecture are also the first five terms of A111151. - Omar E. Pol, Mar 01 2022

			a(1) = 1 because 1 can be represented with a single positive perfect power: 1 = 1^2.
a(2) = 2 because 2 can be represented with two (and not fewer) positive perfect powers with different exponents: 2 = 1^2 + 1^3.
a(6) = 3 because 6 can be represented with three (and not fewer) positive perfect powers with different exponents: 6 = 2^2 + 1^3 + 1^4.
a(7) = 4 because 7 can be represented with four (and not fewer) positive perfect powers with different exponents: 7 = 2^2 + 1^3 + 1^4 + 1^5.
a(15) = 5 because 15 can be represented with five (and not fewer) positive perfect powers with different exponents: 15 = 2^2 + 2^3 + 1^4 + 1^5 + 1^6.

Cf. A111151, A351062, A351063, A351066.

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

A135911 Number of 4-tuples (x,y,z,t) of nonnegative integers such that x^2+y^3+z^4+t^5 = n.

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

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A351064 Minimal number of positive perfect powers, with different exponents, whose sum is n (considering only minimal possible exponents for bases equal to 1).

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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}.

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