A279612 -id:A279612 - cp's OEIS frontend

A338139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x^2 + 26y^2 - 11x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.

1, 2, 2, 2, 3, 4, 2, 2, 4, 5, 3, 4, 3, 4, 3, 2, 4, 6, 3, 5, 6, 4, 2, 4, 4, 5, 4, 4, 4, 6, 2, 2, 7, 5, 3, 6, 5, 4, 3, 5, 7, 8, 1, 4, 8, 4, 2, 4, 5, 6, 4, 5, 5, 6, 4, 4, 8, 5, 2, 6, 4, 3, 3, 2, 8, 11, 3, 5, 11, 6, 1, 6, 8, 7, 5, 4, 6, 5, 1, 5, 10, 10, 5, 9, 8, 5, 4, 4, 8, 14, 5, 5, 8, 4, 4, 4, 6, 7, 5, 7

Offset: 1

Zhi-Wei Sun, Oct 12 2020

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, any positive integer n congruent to 1 or 2 modulo 4 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x^2 + 26*y^2 - 11*x*y = 4^k for some nonnegative integer k.
We have verified this for all n = 1..10^8.
See also A337082 for a similar conjecture.

			a(1) = 1, and 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1^2 + 26*0^2 - 11*1*0 = 2^0.
a(43) = 1, and 43 = 1^2 + 1^2 + 4^2 + 5^2 with 1^2 + 26*1^2 - 11*1*1 = 2^4.
a(6547) = 1, and 6547 = 17^2 + 1^2 + 4^2 + 79^2 with 17^2 + 26*1^2 - 11*17*1 = 2^7.
a(11843) = 1, and 11843 = 3^2 + 1^2 + 13^2 + 108^2 with 3^2 + 26*1^2 - 11*3*1 = 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. See also arXiv:1604.06723 [math.NT].
Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

Cf. A000079, A000118, A000290, A279612, A282463, A338094, A338095, A338096, A338103, A338119, A338121.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[n_]:=PQ[n]=n>0&&IntegerQ[Log[2,n]];
tab={};Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&PQ[x^2+26*y^2-11*x*y],r=r+1],{x,0,Sqrt[n]},{y,Boole[x==0],Sqrt[n-x^2]},{z,0,Sqrt[(n-x^2-y^2)/2]}];tab=Append[tab,r],{n,1,100}];tab

A279616 Numbers of the form x^2 + y^2 + z^2 with x + 2y - 2z a power of four (including 4^0 = 1), where x,y,z are nonnegative integers.

Original entry on oeis.org

1, 3, 4, 5, 9, 10, 14, 16, 17, 18, 19, 20, 22, 24, 29, 33, 34, 35, 37, 41, 45, 48, 49, 50, 51, 52, 53, 58, 59, 61, 64, 65, 66, 68, 69, 70, 73, 74, 77, 78, 80, 82, 84, 88, 89, 90, 94, 97, 98, 99, 100, 104, 106, 107, 109, 113, 114, 116, 117, 121, 122, 125, 129, 130, 132, 133, 138, 139, 141, 144

Offset: 1

Zhi-Wei Sun, Dec 15 2016

nonn

Part (i) of the conjecture in A279612 implies that any positive integer can be written as the sum of a square and a term of the current sequence.

It seems that a(n)/n has the limit 2 as n tends to the infinity.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000079, A000290, A271518, A275656, A275675, A275676, A275738, A278560, A279612.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
FP[n_]:=FP[n]=n>0&&IntegerQ[Log[4,n]];
ex={};Do[Do[If[SQ[m-x^2-y^2]&&FP[x+2y-2*Sqrt[m-x^2-y^2]],ex=Append[ex,m];Goto[aa]],{x,0,Sqrt[m]},{y,0,Sqrt[m-x^2]}];Label[aa];Continue,{m,1,144}]

A281945 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 x + y - z are powers of two (including 2^0 = 1).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 02 2017

nonn

65213 is the first positive integer which cannot be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x and x + y + z are powers of two. Though a(44997) = 0, we have

44997 = 128^2 + (-28)^2 + (-98)^2 + 1^2 with 128 = 2^7 and 128 + (-28) + (-98) = 2^1.

			a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 = 2^0 and 1 + 0 - 0 = 2^0.
a(2237) = 1 since 2237 = 8^2 + 29^2 + 36^2 + 6^2 with 8 = 2^3 and 8 + 29 - 36 = 2^0.
a(4397) = 1 since 4397 = 4^2 + 21^2 + 24^2 + 58^2 with 4 = 2^2 and 4 + 21 - 24 = 2^0.
a(5853) = 1 since 5853 = 2^2 + 52^2 + 52^2 + 21^2 with 2 = 2^1 and 2 + 52 - 52 = 2^1.
a(14711) = 1 since 14711 = 1^2 + 18^2 + 15^2 + 119^2 with 1 = 2^0 and 1 + 18 - 15 = 2^2.
a(16797) = 1 since 16797 = 64^2 + 42^2 + 104^2 + 11^2 with 64 = 2^6 and 64 + 42 - 104 = 2^1.
a(17861) = 1 since 17861 = 32^2 + 0^2 + 31^2 + 126^2 with 32 = 2^5 and 32 + 0 - 31 = 2^0.
a(20959) = 1 since 20959 = 2^2 + 109^2 + 95^2 + 7^2 with 2 = 2^1 and 2 + 109 - 95 = 2^4.
a(21799) = 1 since 21799 = 1^2 + 146^2 + 19^2 + 11^2 with 1 = 2^0 and 1 + 146 - 19 = 2^7.
a(24757) = 1 since 24757 = 64^2 + 56^2 + 119^2 + 58^2 with 64 = 2^6 and 64 + 56 - 119 = 2^0.
a(28253) = 1 since 28253 = 2^2 + 3^2 + 4^2 + 168^2 with 2 = 2^1 and 2 + 3 - 4 = 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, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000079, A000118, A000290, A279612, A281939, A281941.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Pow[n_]:=Pow[n]=n>0&&IntegerQ[Log[2,n]];
Do[r=0;Do[If[SQ[n-4^x-y^2-z^2]&&Pow[2^x+y-z],r=r+1],{x,0,Log[4,n]},{y,0,Sqrt[n-4^x]},{z,0,Sqrt[n-4^x-y^2]}];Print[n," ",r];Continue,{n,1,80}]

A299825 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x <= y, x == y (mod 2), and |x+y-z| is a power of 4 (including 4^0 = 1).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 19 2018

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 6, 13, 15, 18, 27, 43, 53, 63, 93, 107, 109, 123, 141, 159, 165, 173, 477, 493, 653, 1005, 16^k*m (k = 0,1,2,... and m = 3, 4, 7, 8).
We have verified that a(n) > 0 for all n = 1..5*10^6.
A weaker version of the conjecture was proved by the author in arXiv:1701.05868.

			a(8) = 1 since 8 = 2^2 + 2^2 + 0^2 + 0^2 with 2 == 2 (mod 2) and 2 + 2 - 0 = 4.
a(13) = 1 since 13 = 0^2 + 2^2 + 3^2 + 0^2 with 0 == 2 (mod 2) and 0 + 2 - 3 = -4^0.
a(109) = 1 since 109 = 2^2 + 4^2 + 5^2 + 8^2 with 2 == 4 (mod 2) and 2 + 4 - 5 = 4^0.
a(123) = 1 since 123 = 1^2 + 3^2 + 8^2 + 7^2 with 1 == 3 (mod 2) and 1 + 3 - 8 = -4.
a(477) = 1 since 477 = 0^2 + 10^2 + 11^2 + 16^2 with 0 == 10 (mod 2) and 0 + 10 - 11 = -4^0.
a(653) = 1 since 653 = 8^2 + 12^2 + 21^2 + 2^2 with 8 == 12 (mod 2) and 8 + 12 - 21 = -4^0.
a(1005) = 1 since 1005 = 0^2 + 10^2 + 11^2 + 28^2 with 0 == 10 (mod 2) and 0 + 10 - 11 = -4^0.

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, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000118, A271518, A279612, A281945, A281976, A282463.

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

A284343 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that 2*x + y - z is either zero or a power of 8 (including 8^0 = 1).

Original entry on oeis.org

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

Offset: 0

Zhi-Wei Sun, Mar 25 2017

nonn

Conjecture: (i) For any c = 1,2,4, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that c*(2*x+y-z) is either zero or a power of eight (including 8^0 = 1).
(ii) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that P(x,y,z,w) is either zero or a power of four (including 4^0 = 1), whenever P(x,y,z,w) is among the polynomials 2*x-y, x+y-z, x-y-z, x+y-2*z, 2*x+y-z, 2*x-y-z, 2*x-2*y-z, x+2*y-3*z, 2*x+2*y-2*z, 2*x+2*y-4*z, 3*x-2*y-z, x+3*y-3*z, 2*x+3*y-3*z, 4*x+2*y-2*z, 8*x+2*y-2*z, 2*(x-y)+z-w, 4*(x-y)+2*(z-w).
Part (i) of the conjecture is stronger than the first part of Conjecture 4.4 in the linked JNT paper (see also A273432).
Modifying the proofs of Theorem 1.1 and Theorem 1.2(i) in the linked JNT paper slightly, we see that for any a = 1,4 and m = 4,5,6 we can write each n = 0,1,2,... as a*x^m + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x is either zero or a power of two (including 2^0 = 1), and that for any b = 1,2 each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that b*(x-y) is either zero or a power of 4 (including 4^0 = 1).
Starts to differ from A273432 at n=197. - R. J. Mathar, May 25 2023

			a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 = 0 and 2*0 + 0 - 0 = 0.
a(5) = 1 since 5 = 1^2 + 0^2 + 2^2 + 0^2 with 0 < 2 and 2*1 + 0 - 2 = 0.
a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1 < 2 and 2*1 + 1 - 2 = 8^0.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 2 = 2 and 2*4 + 2 - 2 = 8.
a(138) = 1 since 138 = 3^2 + 5^2 + 10^2 + 2^2 with 5 < 10 and 2*3 + 5 - 10 = 8^0.
a(1832) = 1 since 1832 = 4^2 + 30^2 + 30^2 + 4^2 with 30 = 30 and 2*4 + 30 - 30 = 8.
a(2976) = 1 since 2976 = 20^2 + 16^2 + 48^2 + 4^2 with 16 < 48 and 2*20 + 16 - 48 = 8.

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. A000079, A000118, A000290, A000302, A000578, A001018, A271518, A273432, A279612.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Pow[n_]:=Pow[n]=n==0||(n>0&&IntegerQ[Log[8,n]]);
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&Pow[2x+y-z],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[(n-x^2)/2]},{z,y,Sqrt[n-x^2-y^2]}];Print[n," ",r],{n,0,80}]

A350021 Number of ways to write n as w^4 + x^2 + y^2 + z^2 with x - y a power of two (including 2^0 = 1).

Original entry on oeis.org

1, 2, 1, 1, 4, 4, 1, 1, 2, 3, 3, 1, 2, 5, 3, 1, 5, 4, 1, 5, 8, 4, 1, 2, 4, 8, 6, 1, 6, 9, 2, 2, 4, 2, 6, 7, 4, 4, 2, 3, 9, 11, 4, 2, 7, 5, 1, 1, 2, 8, 8, 4, 5, 5, 1, 5, 9, 4, 5, 4, 5, 8, 4, 1, 8, 10, 3, 6, 7, 5, 2, 3, 2, 6, 9, 3, 8, 9, 1, 4, 9, 5, 8, 9, 7, 11, 5, 1, 8, 13, 9, 4, 4, 6, 6, 4, 5, 9, 7, 6

Offset: 1

Zhi-Wei Sun, Dec 08 2021

nonn

Conjecture: a(n) > 0 for all n > 0.
This is a new refinement of Lagrange's four-square theorem, and we have verified it for n up to 10^6.
If x - y = 2^k, then x^2 + y^2 = ((x+y)^2 + (2^k)^2)/2 and x + y >= 2^k. So the above conjecture implies the conjecture in A349661.
In his 2017 JNT paper, the author proved that each n = 0,1,2,... can be written as w^4 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers.
In his 2019 IJNT paper, the author proved that any positive integer can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that x - y is a power of two (including 2^0 = 1).

			a(3) = 1 with 3 = 1^4 + 1^2 + 0^2 + 1^2 and 1 - 0 = 2^0.
a(4) = 1 with 4 = 0^4 + 2^2 + 0^2 + 0^2 and  2 - 0 = 2^1.
a(7) = 1 with 7 = 1^4 + 2^2 + 1^2 + 1^2 and 2 - 1 = 2^0.
a(8) = 1 with 8 = 0^4 + 2^2 + 0^2 + 2^2 and 2 - 0 = 2^1.
a(12) = 1 with 12 = 1^4 + 3^2 + 1^2 + 1^2 and 3 - 1 = 2^1.
a(19) = 1 with 19 = 0^4 + 3^2 + 1^2 + 3^2 and 3 - 1 = 2^1.
a(28) = 1 with 28 = 1^4 + 5^2 + 1^2 + 1^2 and 5 - 1 = 2^2.
a(47) = 1 with 47 = 1^4 + 3^2 + 1^2 + 6^2 and 3 - 1 = 2^1.
a(55) = 1 with 55 = 1^4 + 2^2 + 1^2 + 7^2 and 2 - 1 = 2^0.
a(88) = 1 with 88 = 0^4 + 6^2 + 4^2 + 6^2 and 6 - 4 = 2^1.
a(103) = 1 with 103 = 3^4 + 3^2 + 2^2 + 3^2 and 3 - 2 = 2^0.
a(193) = 1 with 193 = 2^4 + 8^2 + 7^2 + 8^2 and 8 - 7 = 2^0.
a(439) = 1 with 439 = 3^4 + 5^2 + 3^2 + 18^2 and 5 - 3 = 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, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Cf. A000079, A000118, A000290, A000583, A279612, A300708, A338094, A349661, A350012.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PowQ[n_]:=PowQ[n]=IntegerQ[Log[2,n]];
tab={};Do[r=0;Do[If[SQ[n-w^4-x^2-y^2]&&PowQ[y-x],r=r+1],{w,0,(n-1)^(1/4)},{x,0,Sqrt[(n-w^4)/2]},{y,x+1,Sqrt[n-w^4-x^2]}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A338139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x^2 + 26*y^2 - 11*x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A279616 Numbers of the form x^2 + y^2 + z^2 with x + 2*y - 2*z a power of four (including 4^0 = 1), where x,y,z are nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A281945 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 x + y - z are powers of two (including 2^0 = 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A299825 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x <= y, x == y (mod 2), and |x+y-z| is a power of 4 (including 4^0 = 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A284343 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y <= z such that 2*x + y - z is either zero or a power of 8 (including 8^0 = 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A350021 Number of ways to write n as w^4 + x^2 + y^2 + z^2 with x - y a power of two (including 2^0 = 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A338139 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x^2 + 26y^2 - 11x*y a power of two (including 2^0 = 1), where x, y, z, w are nonnegative integers with z <= w.

A279616 Numbers of the form x^2 + y^2 + z^2 with x + 2y - 2z a power of four (including 4^0 = 1), where x,y,z are nonnegative integers.