A281976 -id:A281976 - cp's OEIS frontend

A300510 Number of ways to write n^2 as 4^k*(m^2+1) + x^2 + y^2, where m is 1 or 2, and k,x,y are nonnegative integers with x <= y.

0, 1, 2, 1, 3, 3, 3, 1, 4, 4, 5, 3, 5, 4, 6, 1, 3, 4, 5, 4, 7, 6, 5, 3, 8, 6, 6, 4, 5, 7, 7, 1, 5, 4, 11, 4, 7, 5, 6, 4, 6, 8, 5, 6, 12, 5, 5, 3, 6, 9, 8, 6, 7, 6, 10, 4, 7, 7, 6, 7, 5, 9, 9, 1, 8, 5, 10, 4, 9, 11, 9, 4, 11, 7, 12, 5, 8, 7, 7, 4

Offset: 1

Zhi-Wei Sun, Mar 07 2018

nonn

Conjecture: a(n) > 0 for all n > 1; in other words, for any integer n > 1 there is a nonnegative integer k such that either n^2 - 2*4^k or n^2 - 5*4^k can be written as the sum of two squares. Moreover, a(n) = 1 only for n = 2^k with k > 0.
This conjecture is stronger than the first conjecture in A300448. We have verified that a(n) > 0 for all n = 2..5*10^7.
Consider positive integers c not divisible by 4 such that for any integer n > 1 there is a nonnegative integer k for which n^2 - 2*4^k or n^2 - c*4^k can be written as the sum of two squares. Our computation for n up to 3*10^7 shows that the only candidates for values of c smaller than 160 are 5, 17, 18, 26, 29, 41, 45, 65, 74, 89, 98, 101, 113, 122, 125, 146, 149, 153. These numbers have the form 9^a*(3*b+2) with a and b nonnegative integers and the p-adic order of 3*b+2 is even for any prime p == 3 (mod 4). For n = 42211965 there is no nonnegative integer k such that n^2 - 2*4^k or n^2 - 162*4^k can be written as the sum of two squares.
Qing-Hu Hou at Tianjin Univ. reported that he had verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Mar 14 2018
Qing-Hu Hou found that 29, 65, 113 should be excluded from the candidates. In fact, for c = 29, 65, 113 there is no nonnegative integer k such that N(c)^2 - 2*4^k or N(c)^2 - c*4^k can be written as the sum of two squares, where N(29) = 51883659, N(65) = 56173837 and N(113) = 65525725. - Zhi-Wei Sun, Mar 23 2018
a(n) > 0 for 1 < n < 6*10^9. - Giovanni Resta, Jun 14 2019

			a(1) = 0 since 1^2 - 4^k*(m^2+1) < 0 for k = 0,1,2,... and m = 1, 2.
a(2) = 1 since 2^2 = 4^0*(1^2+1) + 1^2 + 1^2.
a(3) = 2 since 3^2 = 4^0*(2^2+1) + 0^2 + 2^2 = 4^1*(1^2+1) + 0^2 + 1^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000118, A000302, A001481, A271518, A281976, A299924, A299537, A299794, A300362, A300396, A300448, A301452, A301471.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[QQ[n^2-4^k*(m^2+1)],Do[If[SQ[n^2-4^k(m^2+1)-x^2],r=r+1],{x,0,Sqrt[(n^2-4^k(m^2+1))/2]}]],{m,1,2},{k,0,Log[4,n^2/(m^2+1)]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A302981 Number of ways to write n as x^2 + 2*y^2 + 2^z + 2^w, where x,y,z,w are nonnegative integers with z <= w.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 5, 5, 6, 7, 8, 7, 8, 6, 6, 7, 9, 8, 11, 12, 9, 7, 10, 8, 11, 11, 11, 10, 9, 6, 8, 10, 11, 14, 16, 12, 12, 11, 12, 13, 17, 13, 13, 13, 10, 7, 11, 12, 13, 15, 15, 14, 14, 8, 15, 14, 13, 15, 16

Offset: 1

Zhi-Wei Sun, Apr 16 2018

nonn

Clearly, a(2*n) > 0 if a(n) > 0. We note that 52603423 is the first value of n > 1 with a(n) = 0.

See also A302982 and A302983 for related things.

			a(2) = 1 with 2 = 0^2 + 2*0^2 + 2^0 + 2^0.
a(3) = 2 with 3 = 1^2 + 2*0^2 + 2^0 + 2^0 = 0^2 + 2*0^2 + 2^0 + 2^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302982, A302983.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{5,7},Mod[Part[Part[f[n],i],1],8]]&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[n-2^k-2^j],Do[If[SQ[n-2^k-2^j-2x^2],r=r+1],{x,0,Sqrt[(n-2^k-2^j)/2]}]],{k,0,Log[2,n]-1},{j,k,Log[2,Max[1,n-2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]

A302983 Number of ways to write n as x^2 + 2y^2 + 2^z + 32^w with x,y,z,w nonnegative integers.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 5, 4, 5, 6, 4, 8, 8, 7, 12, 8, 6, 9, 9, 6, 13, 13, 8, 13, 12, 8, 13, 14, 11, 15, 17, 8, 14, 11, 11, 16, 17, 11, 17, 19, 8, 17, 19, 10, 19, 18, 12, 15, 17, 12, 20, 17, 13, 20, 18, 16, 24, 18, 15

Offset: 1

Zhi-Wei Sun, Apr 16 2018

nonn

Conjecture: a(n) > 0 for all n > 3.
Clearly, a(2*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 4..6*10^9.
See also A302982 and A302984 for similar conjectures.

			a(4) = 1 with 4 = 0^2 + 2*0^2 + 2^0 + 3*2^0.
a(5) = 2 with 5 = 1^2 + 2*0^2 + 2^0 + 3*2^0 = 0^2 + 2*0^2 + 2^1 + 3*2^0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302984, A302985.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{5,7},Mod[Part[Part[f[n],i],1],8]]&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[n-3*2^k-2^j],Do[If[SQ[n-3*2^k-2^j-2x^2],r=r+1],{x,0,Sqrt[(n-3*2^k-2^j)/2]}]],{k,0,Log[2,n/3]},{j,0,Log[2,Max[1,n-3*2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]

A300751 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 2*x or y or z is a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 11 2018

nonn

Conjecture: a(n) > 0 for all n = 1,2,3,..., and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 0, 1, 4, 5, 6, 7, 8, 20, 24, 28, 31, 36, 43, 61, 71, 79, 100, 116, 157, 188, 200, 344, 351, 388, 632.
This is stronger than the author's 1-3-5 conjecture in A271518. See also A300752 for a similar conjecture stronger than the 1-3-5 conjecture.
a(n) > 0 for all n = 1..3*10^6. - Zhi-Wei Sun, Oct 06 2020

			a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 2*0 = 0^2 and 0 + 3*2 + 5*2 = 4^2.
a(61) = 1 since 61 = 0^2 + 0^2 + 5^2 + 6^2 with 0 = 0^2 and 0 + 3*0 + 5*5 = 5^2.
a(79) = 1 since 79 = 5^2 + 2^2 + 1^2 + 7^2 with 1 = 1^2 and 5 + 3*2 + 5*1 = 4^2.
a(188) = 1 since 188 = 7^2 + 9^2 + 3^2 + 7^2 with 9 = 3^2 and 7 + 3*9 + 5*3 = 7^2.
a(200) = 0 since 200 = 6^2 + 10^2 + 0^2 + 8^2 with 0 = 0^2 and 6 + 3*10 + 5*0 = 6^2.
a(632) = 1 since 632 = 6^2 + 16^2 + 18^2 + 4^2 with 16 = 4^2 and 6 + 3*16 + 5*18 = 12^2.
a(808) = 3 since 808 = 8^2 + 2^2 + 26^2 + 8^2 = 8^2 + 22^2 + 14^2 + 8^2 = 18^2 + 12^2 + 18^2 + 4^2  with 2*8 = 4^2, 2*18 = 6^2 and 8 + 3*2 + 5*26 = 8 + 3*22 + 5*14 = 18 + 3*12 + 5*18 = 12^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000118, A000290, A271518, A281976, A282542, A300666, A300667, A300708, A300712, A300752.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[2(m^2-3y-5z)]||SQ[y]||SQ[z])&&SQ[n-(m^2-3y-5z)^2-y^2-z^2],r=r+1],{m,1,(35n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Min[(m^2-3y)/5,Sqrt[n-y^2]]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A300752 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 11 2018

nonn

Conjecture: a(n) > 0 for all n = 1,2,3,....
This is stronger than the author's 1-3-5 conjecture in A271518. See also A300751 for a similar conjecture stronger than the 1-3-5 conjecture.
In 2020, A. Machiavelo, R. Reis and N. Tsopanidis verified a(n) > 0 for n up to 1.05*10^11. - Zhi-Wei Sun, Oct 06 2020

			a(71) = 1 since 71 = 3^2 + 1^2 + 6^2 + 5^2 with 1 = 1^2 and 3 + 3*1 + 5*6 = 6^2.
a(248) = 1 since 248 = 10^2 + 2^2 + 0^2 + 12^2 with 0 = 0^2 and 10 + 3*2 + 5*0 = 4^2.
a(263) = 1 since 263 = 13^2 + 2^2 + 9^2 + 3^2 with 9 = 3^2 and 13 + 3*2 + 5*9 = 8^2.
a(808) = 1 since 808 = 12^2 + 14^2 + 18^2 + 12^2 with 3*12 = 6^2 and 12 + 3*14 + 5*18 = 12^2.
a(1288) = 1 since 1288 = 12^2 + 18^2 + 26^2 + 12^2 with 3*12 = 6^2 and 12 + 3*18 + 5*26 = 14^2.
a(3544) = 1 since 3544 = 14^2 + 34^2 + 16^2 + 44^2 with 16 = 4^2 and 14 + 3*34 + 5*16 = 14^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
António Machiavelo, Rogério Reis, and Nikolaos Tsopanidis, Report on Zhi-Wei Sun's "1-3-5 conjecture" and some of its refinements, arXiv:2005.13526 [math.NT], 2020.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000118, A000290, A271518, A281976, A282542, A300666, A300667, A300708, A300712, A300751.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[3(m^2-3y-5z)]||SQ[y]||SQ[z])&&SQ[n-(m^2-3y-5z)^2-y^2-z^2],r=r+1],{m,1,(35n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Min[(m^2-3y)/5,Sqrt[n-y^2]]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A302985 Number of ordered pairs (x, y) of nonnegative integers such that n - 2^x - 32^y has the form u^2 + 2v^2 with u and v integers.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 5, 4, 5, 6, 4, 7, 7, 7, 10, 7, 6, 8, 8, 6, 11, 10, 8, 10, 11, 8, 11, 12, 11, 12, 14, 8, 10, 9, 11, 11, 14, 11, 12, 14, 8, 12, 15, 10, 14, 13, 12, 11, 14, 12, 17, 13, 13, 15, 15, 16, 17, 13, 15

Offset: 1

Zhi-Wei Sun, Apr 16 2018

nonn

Conjecture: a(n) > 0 for all n > 3.
This is equivalent to the author's conjecture in A302983. We have verified a(n) > 0 for all n = 4...6*10^9.
See also A302982 and A302984 for similar conjectures.

			a(4) = 1 with 4 - 2^0 - 3*2^0 = 0^2 + 2*0^2.
a(5) = 2 with 5 - 2^0 - 3*2^0 = 1^2 + 2*0^2 and 5 - 2^1 - 3*2^0 = 0^2 + 2*0^2.
a(6) = 2 with 6 - 2^0 - 3*2^0 = 0^2 + 2*1^2 and 6 - 2^1 - 3*2^0 = 1^2 + 2*0^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000290, A002479, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984.

  f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{5,7},Mod[Part[Part[f[n],i],1],8]]&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[n-3*2^k-2^j],r=r+1],{k,0,Log[2,n/3]},{j,0,If[3*2^k==n,-1,Log[2,n-3*2^k]]}];tab=Append[tab,r],{n,1,60}];Print[tab]

A281977 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and -7x - 8y + 8z + 16w are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,....
The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified the conjecture for all n = 0..10^6.
See also A281976, A282013 and A282014 for similar conjectures.
Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^8. - Zhi-Wei Sun, Jun 02 2019

			a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0^2 and -7*0 - 8*0 + 8*0 + 16*1 = 4^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 3^2 + 1^2 with 1 = 1^2 and -7*1 - 8*1 + 8*3 + 16*1 = 5^2.
a(17) = 1 since 17 = 1^2 + 0^2 + 4^2 + 0^2 with 1 = 1^2 and -7*1 - 8*0 + 8*4 + 16*0 = 5^2.
a(28) = 1 since 28 = 4^2 + 2^2 + 2^2 + 2^2 with 4 = 2^2 and -7*4 - 8*2 + 8*2 + 16*2 = 2^2.
a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 1 = 1^2 and -7*1 - 8*1 + 8*2 + 16*5 = 9^2.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 4 = 2^2 and -7*4 -8*2 + 8*2 + 16*4 = 6^2.
a(41) = 1 since 41 = 1^2 + 2^2 + 6^2 + 0^2 with 1 = 1^2 and -7*1 - 8*2 + 8*6 + 16*0 = 5^2.
a(49) = 1 since 49 = 0^2 + 6^2 + 2^2 + 3^2 with 0 = 0^2 and -7*0 - 8*6 + 8*2 + 16*3 = 4^2.
a(241) = 1 since 241 = 9^2 + 4^2 + 12^2 + 0^2 with 9 = 3^2 and -7*9 - 8*4 + 8*12 + 16*0 = 1^2.
a(433) = 1 since 433 = 16^2 + 8^2 + 8^2 + 7^2 with 16 = 4^2 and -7*16 - 8*8 + 8*8 + 16*7 = 0^2.
a(1113) = 1 since 1113 = 1^2 + 30^2 + 4^2 + 14^2 with 1 = 1^2 and -7*1 - 8*30 + 8*4 + 16*14 = 3^2.
a(1521) = 1 since 1521 = 0^2 + 22^2 + 14^2 + 29^2 with 0 = 0^2 and -7*0 - 8*22 + 8*14 + 16*29 = 20^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000118, A000290, A270969, A281939, A281941, A281975, A281976, A282013, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[16*Sqrt[n-x^4-y^2-z^2]+8z-8y-7x^2],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A282013 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 49x + 48(y-z) are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 5, 8, 14, 31, 47, 71, 79, 143, 248, 463, 1039).
The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified a(n) > 0 for all n = 0..10^7.
See also A281976, A281977 and A282014 for similar conjectures.
Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Jun 02 2019

			a(5) = 1 since 5 = 1^2 + 0^2 + 0^2 + 2^2 with 1 = 1^2 and 49*1 + 48*(0-0) = 7^2.
a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 0 = 0^2 and 49*0 + 48*(2-2) = 0^2.
a(14) = 1 since 14 = 1^2 + 2^2 + 3^2 + 0^2 with 1 = 1^2 and 49*1 + 48*(2-3) = 1^2.
a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 1 = 1^2 and 49*1 + 48*(1-2) = 1^2.
a(47) = 1 since 47 = 1^2 + 6^2 + 1^2 + 3^2 with 1 = 1^2 and 49*1 + 48*(6-1) = 17^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 6^2 + 3^2 with 1 = 1^2 and 49*1 + 48*(5-6) = 1^2.
a(79) = 1 since 79 = 1^2 + 7^2 + 2^2 + 5^2 with 1 = 1^2 and 49*1 + 48*(7-2) = 17^2.
a(143) = 1 since 143 = 1^2 + 5^2 + 6^2 + 9^2 with 1 = 1^2 and 49*1 + 48*(5-6) = 1^2.
a(248) = 1 since 248 = 4^2 + 6^2 + 0^2 + 14^2 with 4 = 2^2 and 49*4 + 48*(6-0) = 22^2.
a(463) = 1 since 463 = 9^2 + 6^2 + 15^2 + 11^2 with 9 = 3^2 and 49*9 + 48*(6-15) = 3^2.
a(1039) = 1 since 1039 = 1^2 + 22^2 + 23^2 + 5^2 with 1 = 1^2 and 49*1 + 48*(22-23) = 1^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000118, A000290, A270969, A271518, A281939, A281941, A281975, A281976, A281977, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[49x^2+48(y-z)],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A282014 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 121x + 48(y-z) are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 04 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 29, 31, 40, 94, 104, 143, 319, 671).
The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.
We have verified a(n) > 0 for all n = 0..10^7.
See also A281976, A281977 and A282013 for similar conjectures.
Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Jun 02 2019

			a(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(2-2) = 0^2.
a(29) = 1 since 29 = 0^2 + 5^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(5-2) = 12^2.
a(31) = 1 since 31 = 1^2 + 2^2 + 1^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(2-1) = 13^2.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(2-2) = 22^2.
a(94) = 1 since 94 = 0^2 + 6^2 + 3^2 + 7^2 with 0 = 0^2 and 121*0 + 48*(6-3) = 12^2.
a(104) = 1 since 104 = 4^2 + 6^2 + 6^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(6-6) = 22^2.
a(143) = 1 since 143 = 1^2 + 6^2 + 5^2 + 9^2 with 1 = 1^2 and 121*1 + 48*(6-5) = 13^2.
a(319) = 1 since 319 = 1^2 + 17^2 + 2^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(17-2) = 29^2.
a(671) = 1 since 671 = 9^2 + 5^2 + 23^2 + 6^2 with 9 = 3^2 and 121*9 + 48*(5-23) = 15^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000118, A000290, A270969, A271518, A281939, A281941, A281975, A281976, A281977, A282013.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[121x^2+48(y-z)],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,0,80}]

A282463 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers, x == y (mod 2) and z <= w such that both x and x^2 + 62xy + y^2 are squares.

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Feb 16 2017

nonn

Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 3, 43, 723, 1723, 3571, 3911 and 16^k*m (k = 0,1,2,... and m = 7, 23, 31, 71, 79, 143, 303, 1591).
(ii) Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and a*x^2 + b*x*y + c*y^2 are squares, whenever (a,b,c) is among the ordered triples (84,84,1), (16,144,9), (153,36,100), (177,214,9), (249,114,121).
The author has proved that any nonnegative integer can be expressed as the sum of a fourth power and three squares.

			a(7) = 1 since 7 = 1^2 + 1^2 + 1^2 + 2^2 with 1 == 1 (mod 2), 1 = 1^2 and 1^2 + 62*1*1 + 1^2 = 8^2.
a(23) = 1 since 23 = 1^2 + 3^2 + 2^2 + 3^2 with 1 == 1 (mod 3), 1 = 1^2 and 1^2 + 62*1*3 + 3^2 = 14^2.
a(30) = 2 since 30 = 0^2 + 2^2 + 1^2 + 5^2 with 0 == 2 (mod 3), 0 = 0^2 and 0^2 + 62*0*2 + 2^2 = 2^2, and 30 = 1^2 + 3^2 + 2^2 + 4^2 with 1 == 3 (mod 2), 1 = 1^2 and 1^2 + 62*1*3 + 3^2 = 14^2.
a(79) = 1 since 79 = 1^2 + 7^2 + 2^2 + 5^2 with 1 == 7 (mod 2), 1 = 1^2 and 1^2 + 62*1*7 + 7^2 = 22^2.
a(143) = 1 since 143 = 9^2 + 3^2 + 2^2 + 7^2 with 9 == 3 (mod 2), 9 = 3^2 and 9^2 + 62*9*3 + 3^2 = 42^2.
a(303) = 1 since 303 = 1^2 + 3^2 + 2^2 + 17^2 with 1 == 3 (mod 2), 1 = 1^2 and 1^2 + 62*1*3 + 3^2 = 14^2.
a(723) = 1 since 723 = 1^2 + 7^2 + 12^2 + 23^2 with 1 == 7 (mod 2), 1 = 1^2 and 1^2 + 62*1*7 + 7^2 = 22^2.
a(1591) = 1 since 1591 = 9^2 + 9^2 + 23^2 + 30^2 with 9 == 9 (mod 2), 9 = 3^2 and 9^2 + 62*9*9 + 9^2 = 72^2.
a(1723) = 1 since 1723 = 1^2 + 1^2 + 11^2 + 40^2 with 1 == 1 (mod 2), 1 = 1^2 and 1^2 + 62*1*1 + 1^2 = 8^2.
a(3571) = 1 since 3571 = 9^2 + 3^2 + 0^2 + 59^2 with 9 == 3 (mod 2), 9 = 3^2 and 9^2 + 62*9*3 + 3^2 = 42^2.
a(3911) = 1 since 9^2 + 3^2 + 10^2 + 61^2 with 9 == 3 (mod 2), 9 = 3^2 and 9^2 + 62*9*3 + 3^2 = 42^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000118, A000290, A270969, A271518, A281976, A281977, A282013, A282014.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&Mod[x-y,2]==0&&SQ[x^4+62*x^2*y+y^2],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[(n-x^4-y^2)/2]}];Print[n," ",r];Continue,{n,0,80}]

cp's OEIS Frontend

A300510 Number of ways to write n^2 as 4^k*(m^2+1) + x^2 + y^2, where m is 1 or 2, and k,x,y are nonnegative integers with x <= y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A302981 Number of ways to write n as x^2 + 2*y^2 + 2^z + 2^w, where x,y,z,w are nonnegative integers with z <= w.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A302983 Number of ways to write n as x^2 + 2*y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A300751 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3*y + 5*z a positive square, where x,y,z,w are nonnegative integers such that 2*x or y or z is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A300752 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3*y + 5*z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A302985 Number of ordered pairs (x, y) of nonnegative integers such that n - 2^x - 3*2^y has the form u^2 + 2*v^2 with u and v integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A281977 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and -7*x - 8*y + 8*z + 16*w are squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A282013 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 49*x + 48*(y-z) are squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A302983 Number of ways to write n as x^2 + 2y^2 + 2^z + 32^w with x,y,z,w nonnegative integers.

A300751 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 2*x or y or z is a square.

A300752 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 3y + 5z a positive square, where x,y,z,w are nonnegative integers such that 3*x or y or z is a square.

A302985 Number of ordered pairs (x, y) of nonnegative integers such that n - 2^x - 32^y has the form u^2 + 2v^2 with u and v integers.

A281977 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and -7x - 8y + 8z + 16w are squares.

A282013 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 49x + 48(y-z) are squares.

A282014 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 121x + 48(y-z) are squares.

A282463 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers, x == y (mod 2) and z <= w such that both x and x^2 + 62xy + y^2 are squares.