A301376 -id:A301376 - cp's OEIS frontend

A301471 Number of ways to write n^2 as x^2 + 2y^2 + 32^z with x,y,z nonnegative integers.

0, 1, 2, 1, 3, 4, 3, 1, 5, 4, 4, 4, 5, 4, 10, 1, 4, 7, 4, 4, 10, 4, 3, 4, 6, 6, 11, 4, 7, 10, 6, 1, 9, 5, 7, 7, 7, 6, 12, 4, 6, 12, 7, 4, 14, 4, 8, 4, 3, 8, 10, 6, 8, 13, 6, 4, 16, 8, 7, 10, 7, 6, 14, 1, 7, 11, 6, 5, 16, 9, 5, 7, 7, 7, 18, 6, 7, 14, 6, 4

Offset: 1

Zhi-Wei Sun, Mar 21 2018

nonn

Square Conjecture: a(n) > 0 for all n > 1. Moreover, for any integer n > 3 we can write n^2 as x^2 + 2*y^2 + 3*2^z, where x,y,z are nonnegative integers with y even and z > 1.
It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).
See also A301472 for the list of positive integers not of the form x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
If n^2 = x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers, then it is easy to see that x is not divisible by 3.
The Square Conjecture implies that for each n = 1,2,3,... we can write 3*n^2 as x^2 + 2*y^2 + 2^z with x,y,z nonnegative integers. In fact, if (3*n)^2 = u^2 + 2*v^2 + 3*2^z with u,v,z integers and z >= 0, then u^2 == v^2 (mod 3) and thus we may assume u == v (mod 3) without loss of generality, hence 3*n^2 = (u^2+2*v^2)/3 + 2^z = x^2 + 2*y^2 + 2^z with x = (u+2*v)/3 and y = (u-v)/3 integers.
On March 25, 2018 Qing-Hu Hou at Tianjin Univ. finished his verification of the Square Conjecture for n <= 4*10^8. Then I used Hou's program to verify the conjecture for n <= 5*10^9. - Zhi-Wei Sun, Apr 10 2018
I have found a counterexample to the Square Conjecture, namely a(5884015571) = 0. Note that 5884015571 is the product of the three primes 7, 17 and 49445509. - Zhi-Wei Sun, Apr 15 2018

			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(4) = 1 with 4^2 = 2^2 + 2*0^2 + 3*2^2.
a(1131599953) = 1 with 1131599953^2 = 316124933^2 + 2*768304458^2 + 3*2^6.
a(5884015571) = 0 since there are no nonnegative integers x,y,z such that  x^2 + 2*y^2 + 3*2^z = 5884015571^2.

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. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301472, A301479, A301579, A301640, A302641.

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

A301472 Positive integers not of the form x^2 + 2y^2 + 32^z with x,y,z nonnegative integers.

Original entry on oeis.org

1, 2, 77, 154, 157, 173, 285, 308, 311, 314, 317, 346, 383, 397, 477, 493, 509, 557, 570, 616, 621, 634, 692, 701, 717, 727, 733, 757, 766, 794, 797, 877, 909, 954, 957, 986, 997, 1013, 1018, 1069, 1085, 1093, 1111, 1114, 1117, 1181, 1197, 1221, 1232, 1242, 1268, 1277, 1293

Offset: 1

Zhi-Wei Sun, Mar 21 2018

nonn

It might seem that 1 is the only square in this sequence, but 5884015571^2 is also a term of this sequence.
See also A301471 for related information.
It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).

			a(1) = 1 and a(2) = 2 since x^2 + 2*y^2 + 3*2^z > 2 for all x,y,z = 0,1,2,....

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. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301479.

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]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[Do[If[QQ[m-3*2^k],Goto[aa]],{k,0,Log[2,m/3]}];tab=Append[tab,m];Label[aa],{m,1,1293}];Print[tab]

A302920 Number of ways to write prime(n)^2 as x^2 + 2y^2 + 32^z with x,y,z nonnegative integers.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 4, 3, 7, 6, 7, 6, 7, 8, 8, 7, 7, 6, 5, 7, 6, 8, 6, 8, 7, 9, 9, 7, 6, 6, 9, 7, 5, 8, 5, 9, 9, 10, 10, 9, 14, 7, 5, 11, 8, 8, 11, 10, 10, 12, 10, 6, 12, 11, 10, 8, 9, 10, 11, 8, 7, 15, 5, 11, 8, 14, 10, 7, 10

Offset: 1

Zhi-Wei Sun, Apr 15 2018

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, for any prime p there are nonnegative integers x, y and z such that x^2 + 2*y^2 + 3*2^z = p^2.

As mentioned in A301471, for the composite number m = 5884015571 = 7*17*49445509 there are no nonnegative integers x,y,z such that x^2 + 2*y^2 + 3*2^z = m^2.

			a(1) = 1 with prime(1)^2 = 4 = 1^2 + 2*0^2 + 3*2^0.
a(2) = 2 with prime(2)^2 = 9 = 2^2 + 2*1^2 + 3*2^0 = 1^2 + 2*1^2 + 3*2^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..6000
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. A000040, A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472.

p[n_]:=p[n]=Prime[n];
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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[p[n]^2-3*2^k],Do[If[SQ[p[n]^2-3*2^k-2x^2],r=r+1],{x,0,Sqrt[(p[n]^2-3*2^k)/2]}]],{k,0,Log[2,p[n]^2/3]}];tab=Append[tab,r],{n,1,70}];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]

A301452 Number of ways to write n^2 as m4^k + x^2 + 2y^2 with m in the set {2, 3} and k,x,y nonnegative integers.

Original entry on oeis.org

0, 2, 2, 2, 2, 5, 3, 2, 4, 4, 4, 5, 5, 5, 6, 2, 4, 6, 5, 4, 9, 5, 4, 5, 5, 7, 10, 5, 6, 7, 8, 2, 6, 6, 7, 6, 9, 7, 10, 4, 6, 12, 3, 5, 10, 5, 6, 5, 5, 8, 9, 7, 7, 12, 5, 5, 13, 9, 6, 7, 8, 10, 13, 2, 6, 8, 10, 6, 15, 9, 9, 6, 10, 9, 12, 7, 8, 13, 6, 4

Offset: 1

Zhi-Wei Sun, Mar 21 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
We call this the 2-3 conjecture. It is simialr to the author's 2-5 conjecture which states that A300510(n) > 0 for all n > 1.
We have verified that a(n) > 0 for all n = 2..5*10^7.
It is known that the number of ways to write a positive integer n as x^2 + 2*y^2 with x and y integers is twice the difference |{d > 0: d|n and d == 1,3 (mod 8)}| - |{d>0: d|n and d == 5,7 (mod 8)}|.

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

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. A000290, A000302, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391.

f[n_]:=f[n]=n/2^(IntegerExponent[n,2]);
OD[n_]:=OD[n]=Divisors[f[n]];
QQ[n_]:=QQ[n]=(n==0)||(n>0&&Sum[JacobiSymbol[-2,Part[OD[n],i]],{i,1,Length[OD[n]]}]!=0);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[QQ[n^2-m*4^k],Do[If[SQ[n^2-m*4^k-2x^2],r=r+1],{x,0,Sqrt[(n^2-m*4^k)/2]}]],{m,2,3},{k,0,Log[4,n^2/m]}];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]

A301579 Least nonnegative integer k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers, or -1 if no such k exists.

Original entry on oeis.org

-1, 0, 0, 2, 0, 0, 1, 4, 1, 0, 0, 2, 0, 0, 1, 6, 1, 0, 0, 2, 0, 2, 1, 4, 1, 0, 0, 2, 0, 3, 3, 8, 1, 0, 3, 2, 0, 0, 3, 4, 1, 0, 1, 4, 0, 0, 1, 6, 3, 0, 0, 2, 1, 0, 1, 4, 3, 0, 1, 5, 0, 5, 1, 10, 1, 0, 0, 2, 3, 0, 4, 4, 1, 2, 0, 2, 0, 0, 3, 6

Offset: 1

Zhi-Wei Sun, Mar 23 2018

sign

The Square Conjecture in A301471 implies that a(n) >= 0 for all n > 1.
It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).
Numbers t such that a(t) = 0 are 2, 3, 5, 6, 10, 11, 13, 14, 18, 19, 21, 26, 27, 29, 34, 37, ... - Altug Alkan, Mar 26 2018

			a(1) = -1 since 1^2 - 3*2^k < 0 for all k = 0,1,2,....
a(31) = 3 since 31^2 - 3*2^3 = 17^2 + 2*18^2.
a(2^k) = 2*k - 2 for all k = 1,2,3,..., because (2^k)^2 - 3*2^(2*k-2) = (2^(k-1))^2 + 2*0^2, and (2^k)^2 - 3*2^j = 2^j*(2^(2*k-j) - 3) with 0 <= j < 2*k-2 cannot be written as x^2 + 2*y^2 with x and y integers.

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. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A301479.

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[Do[If[QQ[n^2-3*2^k],tab=Append[tab,k];Goto[aa]],{k,0,Log[2,n^2/3]}];tab=Append[tab,-1];Label[aa],{n,1,80}];Print[tab]

A301640 Largest integer k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers, or -1 if no such k exists.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 25 2018

sign

Conjecture: a(n) > 0.6*log_2(log_2 n) for all n > 2, and also lim inf_{n->infinity} a(n)/(log n) = 0.
The author's Square Conjecture in A301471 would imply that a(n) >= 0 for all n > 1. We have verified that a(n) > 0.6*log_2(log_2 n) for all n = 3..4*10^9. For n = 2857932461, we have a(n) = 3 and 0.603 < a(n)/log_2(log_2 n) < 0.604.
It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).

			a(2) = 0 since 2^2 - 3*2^0 = 1^2 + 2*0^2.
a(3) = 1 since 3^2 - 3*2^1 = 2^2 + 2*1^2.
a(5) = 3 since 5^2 - 3*2^3 = 1^2 + 2*0^2.
a(6434567) = 10 since 6434567^2 - 3*2^10 = 5921293^2 + 2*1780722^2.

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. A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A301479, A301579.

f:= proc(n) local k,t;
    for k from floor(log[2](n^2/3)) by -1 to 0 do
       if g(n^2 - 3*2^k) then return k fi
    od;
    -1
end proc:
map(f, [$1..100]); # Robert Israel, Mar 26 2018

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[Do[If[QQ[n^2-3*2^(Floor[Log[2,n^2/3]]-k)],tab=Append[tab,Floor[Log[2,n^2/3]]-k];Goto[aa]],{k,0,Log[2,n^2/3]}];tab=Append[tab,-1];Label[aa],{n,1,70}];Print[tab]

A303235 Number of ordered pairs (x, y) with 0 <= x <= y such that n - 2^x - 2^y can be written as the sum of two triangular numbers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 20 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
Note that a nonnegative integer m is the sum of two triangular numbers if and only if 4*m+1 (or 8*m+2) can be written as the sum of two squares.
We have verified a(n) > 0 for all n = 2..4*10^8. See also the related sequences A303233 and A303234.

			a(2) = 1 with 2 - 2^0 - 2^0 = 0*(0+1)/2 + 0*(0+1)/2.
a(3) = 2 with 3 - 2^0 - 2^0 = 0*(0+1)/2 + 1*(1+1)/2 and 3 - 2^0 - 2^1 = 0*(0+1)/2 + 0*(0+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, 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, A000217, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234.

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]);
tab={};Do[r=0;Do[If[QQ[4(n-2^k-2^j)+1],r=r+1],{k,0,Log[2,n]-1},{j,k,Log[2,n-2^k]}];tab=Append[tab,r],{n,1,60}];Print[tab]

cp's OEIS Frontend

A301471 Number of ways to write n^2 as x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.

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A301472 Positive integers not of the form x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.

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A302920 Number of ways to write prime(n)^2 as x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.

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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.

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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.

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A301452 Number of ways to write n^2 as m*4^k + x^2 + 2*y^2 with m in the set {2, 3} and k,x,y nonnegative integers.

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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.

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A301579 Least nonnegative integer k such that n^2 - 3*2^k can be written as x^2 + 2*y^2 with x and y integers, or -1 if no such k exists.

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A301471 Number of ways to write n^2 as x^2 + 2y^2 + 32^z with x,y,z nonnegative integers.

A301472 Positive integers not of the form x^2 + 2y^2 + 32^z with x,y,z nonnegative integers.

A302920 Number of ways to write prime(n)^2 as x^2 + 2y^2 + 32^z with x,y,z nonnegative integers.

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

A301452 Number of ways to write n^2 as m4^k + x^2 + 2y^2 with m in the set {2, 3} and k,x,y nonnegative integers.

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.

A301579 Least nonnegative integer k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers, or -1 if no such k exists.

A301640 Largest integer k such that n^2 - 32^k can be written as x^2 + 2y^2 with x and y integers, or -1 if no such k exists.