A301471 -id:A301471 - cp's OEIS frontend

A302641 Number of nonnegative integers k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers.

0, 1, 2, 1, 3, 3, 3, 1, 3, 4, 4, 3, 4, 4, 5, 1, 4, 4, 4, 4, 5, 4, 3, 3, 6, 5, 5, 4, 5, 5, 5, 1, 4, 5, 6, 4, 5, 5, 6, 4, 5, 6, 6, 4, 7, 4, 7, 3, 3, 7, 4, 5, 6, 6, 5, 4, 7, 6, 6, 5, 6, 5, 6, 1, 7, 5, 6, 5, 7, 7, 4, 4, 6, 5, 8, 5, 6, 7, 5, 4

Offset: 1

Zhi-Wei Sun, Apr 10 2018

nonn

The author's Square Conjecture in A301471 implies that a(n) > 0 for all n > 1.

We have a(2^n) = 1 for all n > 0. In fact, (2^n)^2 = (2^(n-1))^2 + 2*0^2 + 3*2^(2*n-2). If k > 2*n-2 then 3*2^k >= 6*2^(2*n-2) > (2^n)^2. If 0 <= k < 2*n-2, then 2*n-k is at least 3 and hence (2^n)^2 - 3*2^k = 2^k*(2^(2*n-k)-3) cannot be written as x^2 + 2*y^2 with x and y integers.

			a(2) = 1 with 2^2 = 1^2 + 2*0^2 + 3*2^0.
a(3) = 2 with 3^2 = 2^2 + 2*1^2 + 3*2^0 = 1^2 + 2*1^2 + 3*2^1.
a(2857932461) = 1 since 3 is the only nonnegative integer k such that 2857932461^2 - 3*2^k has the form x^2 + 2*y^2 with x and y integers.
a(4428524981) = 2 since 3 and 8 are the only nonnegative integers k such that 4428524981^2 - 3*2^k has the form x^2 + 2*y^2 with x and y integers.
a(4912451281) = 3 since 3, 6 and 7 are the only nonnegative integers k with 4428524981^2 - 3*2^k = x^2 + 2*y^2 for some integers x and y.

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.
Zhi-Wei Sun, My square conjecture with prize, a message to Number Theory List, April 7, 2018.

Cf. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301472, A301479, A301579, A301640.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[(Mod[Part[Part[f[n],i],1],8]==5||Mod[Part[Part[f[n],i],1],8]==7)&&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^2-3*2^k],r=r+1],{k,0,Log[2,n^2/3]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A301805 Number of ways to write 3n^2 as x^2 + 10y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 27 2018

nonn

It might seem that a(n) > 0 for all n > 2. However, we find that a(323525083) = 0, moreover 3*323525083^2 cannot be written as x^2 + 10*y^2 + 2^z with x,y,z nonnegative integers. We note also that a(270035155) = 0 but 3*270035155^2 - 2^0 has the form x^2 + 10*y^2 with x and y integers.
My way to check whether 3*n^2 can be written as x^2 + 10*y^2 + 2^z is to find z such that 3*n^2 - 2^z can be written as x^2 + 10*y^2. I observe that a positive integer n has the form x^2 + 10*y^2 with x and y integers if and only if the p-adic order ord_p(n) of n is even for any prime p == 3, 17, 21, 27, 29, 31, 33, 39 (mod 40) and the sum of those ord_p(n) with p prime and p == 2, 5, 7, 13, 23, 37 (mod 40) is even.
From David A. Corneth, Mar 27 2018: (Start)
If a(n) > 0 then a(2*n) > 0; 3*n^2 = x^2 + 10*y^2 + 2^z <=> 3*(2*n)^2 = 4 * 3*n^2 = 4 * (x^2 + 10*y^2 + 2^z) = (2*x)^2 + 10 * (2*y)^2 + 2^(z + 2).
So we just need to check odd n and as z > 0, 2 | 2^z and furthermore 2 | 10 * y^2 so 3*x^2 must be odd, i.e., x must be odd for 3*n^2 to be odd. Also, y must be odd. For odd n, 3*n^2 == 3 (mod 4), for odd x, x^2 == (1 mod 4), for z >= 3, 2^z == 0 (mod 4) so 10 * y^2 must be == 2 (mod 4) which happens if and only if y is odd. (End)

			a(1) = a(2) = 0 since 3*1^2 < 3*2^2 < 2^4.
a(3) = 1 since 3*3^2 = 1^2 + 10*1^2 + 2^4.
a(4) = 1 since 3*4^2 = 4^2 + 10*0^2 + 2^5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..800
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. A000079, A000290, A020673, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A301479, A301579, A301640.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[3*n^2-2^k-10x^2],r=r+1],{k,4,Log[2,3n^2]},{x,0,(3*n^2-2^k)/10}];tab=Append[tab,r],{n,1,80}];Print[tab]

A301858 Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.

Original entry on oeis.org

1, 5, 29, 65

Offset: 1

Zhi-Wei Sun, Mar 27 2018

nonn

The sequence has no term in the interval [66, 10^6].
Conjecture 1: The sequence only has the four terms 1, 5, 29 and 65.
Conjecture 2: For any integer n > 1 which is neither 17 nor a power of 2, if n = u^2 + 2*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with z nonzero.
Conjecture 3: For any positive integer n not of the form 4^k*m (k = 0,1,2,... and m = 1, 7, 13), if n = u^2 + 3*v^2 for some integers u and v, then n = x^2 + 2*y^2 + 3*z^2 for some integers x,y,z with y nonzero.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4

Cf. A000290, A000549, A001481, A002479, A301471.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&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[If[QQ[m]==False,Goto[aa]];Do[If[SQ[m-2x^2-y^2],Goto[aa]],{x,1,Sqrt[m/2]},{y,0,Sqrt[(m-2x^2)/2]}];tab=Append[tab,m];Label[aa],{m,1,1000}];Print[tab]

A301891 Number of ways to write 2n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3y + 5z + 15w is a power of 4.

Original entry on oeis.org

2, 1, 1, 2, 3, 1, 2, 1, 3, 1, 2, 1, 6, 3, 8, 2, 5, 4, 6, 3, 4, 3, 6, 1, 3, 3, 7, 2, 8, 9, 8, 1, 15, 5, 8, 3, 11, 1, 5, 1, 4, 4, 2, 2, 10, 7, 17, 1, 18, 11, 14, 6, 16, 6, 17, 3, 21, 10, 16, 8, 19, 8, 30, 2, 15, 9, 18, 5, 28, 5, 27, 4, 13, 11, 24, 6, 28, 17, 20, 3

Offset: 1

Zhi-Wei Sun, Mar 28 2018

nonn

Conjecture 1: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m with k = 0,1,2,... and m = 2, 3, 6, 10, 38.

Conjecture 2: For any positive integer n, we can write 2*n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3*y + 5*z + 15*w is twice a power of 4.

			a(1) = 2 since 2*1^2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 + 3*1 + 5*0 + 15*0 = 4, and 2*1^2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 3*0 + 5*0 + 15*1 = 4^2.
a(2) = 1 since 2*2^2 = 0^2 + 2^2 + 2^2 + 0^2 with 0 + 3*2 + 5*2 + 15*0 = 4^2.
a(3) = 1 since 2*3^2 = 1^2 + 1^2 + 0^2 + 4^2 with 1 + 3*1 + 5*0 + 15*4 = 4^3.
a(6) = 1 since 2*6^2 = 0^2 + 8^2 + 2^2 + 2^2 with 0 + 3*8 + 5*2 + 15*2 = 4^3.
a(10) = 1 since 2*10^2 = 10^2 + 8^2 + 6^2 + 0^2 with 10 + 3*8 + 5*6 + 15*0 = 4^3.
a(38) = 1 since 2*38^2 = 34^2 + 34^2 + 24^2 + 0^2 with 34 + 3*34 + 5*24 + 15*0 = 4^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..225
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. A000290, A000302, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301579, A301640.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Pow[n_]:=Pow[n]=IntegerQ[Log[4,n]]
tab={};Do[r=0;Do[If[SQ[2n^2-x^2-y^2-z^2]&&Pow[x+3y+5z+15*Sqrt[2n^2-x^2-y^2-z^2]],r=r+1],{x,0,Sqrt[2]n},{y,0,Sqrt[2n^2-x^2]},{z,0,Sqrt[2n^2-x^2-y^2]}];tab=Append[tab,r],{n,1,80}];Print[tab]

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A302641 Number of nonnegative integers k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers.

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A301805 Number of ways to write 3n^2 as x^2 + 10y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.

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A301858 Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.

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A301891 Number of ways to write 2n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3y + 5z + 15w is a power of 4.

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cp's OEIS Frontend

A302641 Number of nonnegative integers k such that n^2 - 3*2^k can be written as x^2 + 2*y^2 with x and y integers.

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A301805 Number of ways to write 3*n^2 as x^2 + 10*y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.

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A301858 Positive integers which can be written as the sum of two squares but cannot be written as x^2 + y^2 + 2*z^2 with x and y integers and z a nonzero integer.

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A301891 Number of ways to write 2*n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3*y + 5*z + 15*w is a power of 4.

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A302641 Number of nonnegative integers k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers.

A301805 Number of ways to write 3n^2 as x^2 + 10y^2 + 2^z, where x, y and z are nonnegative integers with z > 3.

A301891 Number of ways to write 2n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3y + 5z + 15w is a power of 4.