A274274 -id:A274274 - cp's OEIS frontend

A275169 Positive integers not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

15, 21, 47, 53, 79, 85, 92, 111, 117, 120, 181, 183, 245, 309, 311, 335, 372, 373, 398, 405, 421, 437, 447, 501, 565, 573, 629, 636, 645, 655, 693, 757, 791, 807, 820, 821, 853, 869, 885, 888, 949, 967, 1013, 1045, 1077, 1141, 1205, 1223, 1269, 1271, 1303, 1461, 1555, 1591, 1613, 1653, 2087, 2101, 2255, 2421

Offset: 1

Zhi-Wei Sun, Jul 18 2016

nonn

Conjecture: The sequence has totally 174 terms as listed in the b-file the largest of which is 375565.
This implies the conjecture in A275150. We note that the sequence contains no term greater than 375565 and not exceeding 10^6.
See also A275168 for a similar conjecture.

			a(1) = 15 since 15 is the least positive integer not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

Zhi-Wei Sun, Table of n, a(n) for n = 1..174

Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275168.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[m-x^3-2*y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/2]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,2421}]

A275083 Positive integers congruent to 0 or 1 modulo 4 that cannot be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

Original entry on oeis.org

120, 312, 813, 2136, 2680, 3224, 4404, 5340, 6420, 10060, 11320, 11824, 14008, 15856, 26544, 28804, 34392, 47984

Offset: 1

Zhi-Wei Sun, Jul 15 2016

nonn

Conjecture: (i) The sequence has totally 18 terms as listed.
(ii) For each r = 2,3 there are infinitely many positive integers n == r (mod 4) not in the form x^3 + y^2 + z^2 with x,y,z nonnegative integers.
Our computation indicates that the sequence has no other terms below 10^6.
Let d be 2 or 6. Clearly, n-d is congruent to 0 or 1 modulo 4 if n is congruent to 2 or 3 modulo 4. So part (i) of the conjecture essentially implies that for each n = 0,1,2,... either n or n-d can be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

			a(1) = 120 since all those positive integers congruent to 0 or 1 modulo 4 and smaller than 120 can be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers but 120 (divisible by 4) cannot be written in this way.

Cf. A000290, A000578, A022551, A022552, A262813, A262857, A272979, A274274.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[If[Mod[m,4]>1,Goto[aa]];Do[If[SQ[m-x^3-y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/2]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,50000}]

A275150 Number of ordered ways to write n as x^3 + 2y^2 + kz^2, where x,y,z are nonnegative integers, k is 1 or 5, and k = 1 if z = 0.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 1, 2, 4, 3, 4, 3, 2, 2, 3, 3, 3, 3, 4, 5, 2, 3, 2, 3, 5, 4, 4, 5, 3, 4, 3, 2, 3, 2, 2, 5, 5, 4, 2, 2, 5, 3, 5, 5, 3, 5, 5, 2, 3, 3, 4, 4, 2, 2, 4, 4, 6, 3, 5, 4, 2, 3, 4, 5, 5, 4, 4, 5, 5, 5, 1, 5

Offset: 0

Zhi-Wei Sun, Jul 17 2016

nonn

Conjecture 1: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 15, 79, 120, 218, 399, 454, 622, 725, 3240.
We have verified that a(n) > 0 for all n = 0..10^7.
Conjecture 2: For any positive integers a, b, c and integers i, j, k greater than one, there are infinitely many positive integers not in the set {a*x^i + b*y^j + c*z^k: x,y,z = 0,1,2,...}. - Zhi-Wei Sun, May 24 2023

			a(0) = 1 since 0 = 0^3 + 2*0^2 + 0^2.
a(15) = 1 since 15 = 2^3 + 2*1^2 + 5*1^2.
a(79) = 1 since 79 = 3^3 + 2*4^2 + 5*2^2.
a(120) = 1 since 120 = 2^3 + 2*4^2 + 5*4^2.
a(218) = 1 since 218 = 6^3 + 2*1^2 + 0^2.
a(399) = 1 since 399 = 5^3 + 2*3^2 + 16^2.
a(454) = 1 since 454 = 0^3 + 2*15^2 + 2^2.
a(622) = 1 since 622 = 2^3 + 2*17^2 + 6^2.
a(725) = 1 since 725 = 5^3 + 2*10^2 + 20^2.
a(3240) = 1 since 3240 = 7^3 + 2*38^2 + 3^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
G. Doyle and K. S. Williams, A positive-definite ternary quadratic form does not represent all positive integers, Integers 17 (2017), #A41, 19pp (electronic).
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000290, A000578, A262813, A262941, A262954, A270488, A274274, A275083.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
TQ[n_]:=TQ[n]=SQ[n]||SQ[n/5]
Do[r=0;Do[If[TQ[n-x^3-2*y^2],r=r+1],{x,0,n^(1/3)},{y,0,Sqrt[(n-x^3)/2]}];Print[n," ",r];Continue,{n,0,80}]

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

Original entry on oeis.org

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}]

A297788 Number of partitions of n into 3 squares and a nonnegative cube.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 1, 2, 4, 4, 3, 3, 3, 3, 1, 2, 5, 5, 4, 3, 3, 3, 1, 2, 5, 6, 6, 4, 4, 5, 2, 3, 6, 6, 6, 5, 6, 5, 3, 3, 7, 6, 4, 6, 6, 6, 2, 3, 7, 6, 7, 6, 7, 8, 3, 4, 6, 6, 6, 5, 6, 8, 4, 4, 9, 8, 8, 7, 8, 7, 2, 6, 10, 9, 8, 8, 9, 7, 2, 6, 12, 11, 8, 7, 7

Offset: 0

XU Pingya, Jan 06 2018

nonn

When n is not of the form 4^a * (8b + 7), according to Legendre's three-square theorem, n = x^2 + y^2 + z^2 = x^2 + y^2 + z^2 + 0^3 (where a, b, x, y and z are nonnegative integers with x <= y <= z).
If n = 8b + 7, then n - 1 = 8b + 6 is not of the form 4^a * (8b + 7). So n = (n - 1) + 1 = x^2 + y^2 + z^2 + 1^3.
If n = 4 * (8b + 7), then n - 1 = 8 * (4b + 3) + 3 is also not of the form 4^a * (8b + 7).
If n = 4^2 * (8b + 7), then n - 8 = 4 * (8 * (4b + 3) + 2) is not of the form 4^a * (8b + 7).  n = (n - 8) + 8 = x^2 + y^2 + z^2 + 2^3.
If n = 4^k * (8b + 7) (k >= 3), then n - 8 = 4 * (8 * (4^(k - 1) * b + 4^(k - 3) * 14) - 2) = 4 * (8m - 2) is also not of the form 4^a * (8b + 7).
That is, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative cube, so a(n) > 0.

			2 = 0^2 + 0^2 + 1^2 + 1^3 = 0^2 + 1^2 + 0^2 + 1^3, a(2) = 2.
9 = 0^2 + 0^2 + 1^2 + 2^3 = 0^2 + 1^2 + 0^2 + 2^3 = 0^2 + 2^2 + 2^2 + 1^3 = 1^2 + 2^2 + 2^2 + 0^3, a(9) = 4.

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Legendre's three-square theorem

Cf. A002635, A004215, A274274.

N:= 100: # to get a(0)..a(N)
A:= Array(0..N):
for x from 0 to floor(sqrt(N)) do
  for y from 0 to x while x^2 + y^2 <= N do
    for z from 0 to y while x^2 + y^2 + z^2 <= N do
      for w from 0 do
        t:= x^2 + y^2 + z^2 + w^3;
        if t > N then break fi;
        A[t]:= A[t]+1;
od od od od:
convert(A,list); # Robert Israel, Jan 11 2018

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

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}]

cp's OEIS Frontend

A275169 Positive integers not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

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A275083 Positive integers congruent to 0 or 1 modulo 4 that cannot be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

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A275150 Number of ordered ways to write n as x^3 + 2*y^2 + k*z^2, where x,y,z are nonnegative integers, k is 1 or 5, and k = 1 if z = 0.

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

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A297788 Number of partitions of n into 3 squares and a nonnegative cube.

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

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A275150 Number of ordered ways to write n as x^3 + 2y^2 + kz^2, where x,y,z are nonnegative integers, k is 1 or 5, and k = 1 if z = 0.