A275738 -id:A275738 - cp's OEIS frontend

A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2y - 2z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

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

Offset: 1

Zhi-Wei Sun, Dec 15 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 16^k*q (k = 0,1,2,... and q = 1, 2, 3, 6, 7, 8, 12, 15, 27, 31, 47, 72, 76, 92, 111, 127).
(ii) Let a and b be positive integers with gcd(a,b) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x - b*y a power of two (including 2^0 = 1) if and only if (a,b) = (1,1), (2,1), (2,3).
(iii) Let a,b,c be positive integers with a <= b and gcd(a,b,c) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y - c*z a power of two if and only if (a,b,c) is among the triples (1,1,1), (1,1,2), (1,2,1), (1,2,2), (1,3,1), (1,3,2), (1,3,3), (1,3,4), (1,3,5), (1,4,1), (1,4,2), (1,4,3), (1,4,4), (1,5,1), (1,5,2), (1,5,4), (1,5,5,), (1,6,3), (1,7,4), (1,7,7), (1,8,1), (1,9,2), (2,3,1), (2,3,3), (2,3,4), (2,5,1), (2,5,3), (2,5,4), (2,5,5), (2,7,1), (2,7,3), (2,7,7), (2,9,3), (2,11,5), (3,4,3), (7,8,7).
(iv) Let a,b,c be positive integers with b <= c and gcd(a,b,c) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x - b*y - c*z a power of two if and only if (a,b,c) is among the triples (2,2,1), (4,2,1), (4,3,1), (4,4,3).
(v) Let a,b,c be positive integers with a <= b, c <= d, and gcd(a,b,c,d) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y - c*z -d*w a power of two if and only if (a,b,c,d) is among the quadruples (1,2,1,1), (1,2,1,2), (1,2,1,3), (1,3,1,2), (1,3,2,3), (1,3,2,4), (1,4,1,2), (1,7,2,6), (1,9,1,4), (2,2,2,3), (2,3,1,2), (2,3,1,3), (2,3,2,3), (2,3,6,1), (2,4,1,2), (2,5,1,2), (2,5,2,3), (2,5,3,4), (3,4,1,2), (3,4,1,3), (3,4,1,5), (3,4,2,5), (3,4,3,4), (3,8,1,10), (3,8,2,3), (4,5,1,5).
(vi) Let a,b,c be positive integers with a <= b <= c and gcd(a,b,c,d) odd. Then any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y + c*z -d*w a power of two if and only if (a,b,c,d) is among the quadruples (1,1,2,2), (1,1,2,3), (1,1,2,4), (1,2,2,3), (1,2,3,4), (1,2,4,3), (1,2,6,7), (1,3,4,4), (1,4,6,5), (2,3,5,4).
(vii) For any positive integers a,b,c,d, not all positive integers can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x - b*y - c*z -d*w a power of two.
(viii) Let a and b be positive integers, and c and d be nonnegative integers. Then, not all positive integers can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and a*x + b*y + c*z + d*w a power of two.
We have verified a(n) > 0 for all n = 1..2*10^7. The conjecture that a(n) > 0 for all n > 0 appeared in arXiv:1701.05868.

			a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 2*1 - 2*1 = 4^0.
a(15) = 1 since 15 = 3^2 + 1^2 + 2^2 + 1^2 with 3 + 2*1 - 2*2 = 4^0.
a(27) = 1 since 27 = 4^2 + 1^2 + 1^2 + 3^2 with 4 + 2*1 - 2*1 = 4.
a(31) = 1 since 31 = 3^2 + 2^2 + 3^2 + 3^2 with 3 + 2*2 - 2*3 = 4^0.
a(47) = 1 since 47 = 3^2 + 2^2 + 3^2 + 5^2 with 3 + 2*2 - 2*3 = 4^0.
a(72) = 1 since 72 = 8^2 + 0^2 + 2^2 + 2^2 with 8 + 2*0 - 2*2 = 4.
a(76) = 1 since 76 = 1^2 + 5^2 + 5^2 + 5^2 with 1 + 2*5 - 2*5 = 4^0.
a(92) = 1 since 92 = 4^2 + 6^2 + 6^2 + 2^2 with 4 + 2*6 - 2*6 = 4.
a(111) = 1 since 111 = 9^2 + 1^2 + 5^2 + 2^2 with 9 + 2*1 - 2*5 = 4^0.
a(127) = 1 since 127 = 7^2 + 2^2 + 5^2 + 7^2 with 7 + 2*2 - 2*5 = 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. Also available from arXiv:1604.06723 [math.NT].
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000079, A000118, A000290, A271518, A275656, A275675, A275676, A275738, A278560, A279616.

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

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}]

A275471 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+z^2, where k,x,y,z are nonnegative integers with x <= y and x == y (mod 2).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Aug 11 2016

nonn

Conjecture: a(n) > 0 except for n = 449.
See also A275656, A275678 and A275738 for related conjectures.
As x^2 + y^2 = 2*((x+y)/2)^2 + 2*((x-y)/2)^2, we see that {x^2 + y^2: x and y are integers with x == y (mod 2)} = {2*x^2 + 2*y^2: x and y are integers}.

			a(8) = 1 since 8 = 4*(1+0^2+0^2) + 2^2 with 0+0 even.
a(31) = 1 since 31 = 4^0*(1+1^2+5^2) + 2^2 with 1+5 even.
a(47) = 1 since 47 = 4^0*(1+1^2+3^2) + 6^2 with 1+3 even.
a(79) = 1 since 79 = 4^0*(1+5^2+7^2)+2^2 with 5+7 even.
a(1009) = 1 since 1009 = 4^2*(1+1^2+1^2) + 31^2 with 1+1 even.
a(7793) = 1 since 7793 = 4^2*(1+12^2+18^2) + 17^2 with 12+18 even.

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. A000118, A000290, A271518, A275648, A275656, A275675, A275676, A275678, A275738.

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

cp's OEIS Frontend

A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2y - 2z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

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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.

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A275471 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+z^2, where k,x,y,z are nonnegative integers with x <= y and x == y (mod 2).

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cp's OEIS Frontend

A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2*y - 2*z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

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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.

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A275471 Number of ordered ways to write n as 4^k*(1+x^2+y^2)+z^2, where k,x,y,z are nonnegative integers with x <= y and x == y (mod 2).

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A279612 Number of ways to write n = x^2 + y^2 + z^2 + w^2 with x + 2y - 2z a power of 4 (including 4^0 = 1), where x,y,z,w are nonnegative integers.

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.