A296523 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x > 0, y >= 0 and 0 <= z <= w such that 64*x^2 + 65*y^2 = q^2 for some practical number q.

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

Offset

1,5

Comments

Conjecture: (i) a(n) > 0 for all n > 0. Also, a(n) = 1 only for those numbers 2^k (k = 0,1,2,...), 4^k*m (k = 0,1,...; m = 3, 7), 4^k*47 (k = 0,1,2,3), 4^k*s (k = 0,1,2; s = 15, 23, 31, 39, 71); 4^k*t (k =0,1; t = 87, 111, 119, 159, 191, 311).

(ii) Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^2 + b*y^2 = q^2 for some practical number q, provided that (a,b) is among the ordered pairs (5,16), (7,36), (16,77), (36,55), (36,91), (36, -5), (64,-7).

Links

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.

Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT].)

Example

a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 and 64*1^2 + 65*0^2 = 8^2 with 8 practical.
a(71) = 1 since 71 = 3^2 + 6^2 + 1^2 + 5^2 and 64*3^2 + 65*6^2 = 54^2 with 54 practical.
a(159) = 1 since 159 = 5^2 + 10^2 + 3^2 + 5^2 and 64*5^2 + 65*10^2 = 90^2 with 90 practical.
a(311) = 1 since 311 = 1^2 + 2^2 + 9^2 + 15^2 and 64*1^2 + 65*2^2 = 18^2 with 18 practical.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
pQ[n_]:=pQ[n]=SQ[n]&&pr[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&pQ[64x^2+65y^2], r=r+1], {x, 1, Sqrt[n]}, {y, 0, Sqrt[n-x^2]}, {z, 0, Sqrt[(n-x^2-y^2)/2]}]; tab=Append[tab, r], {n, 1, 70}]

Crossrefs

Cf. A000118, A000290, A271518, A281976.

Sequence in context: A109439 A247646 A133333 * A171876 A306462 A133332

Adjacent sequences: A296520 A296521 A296522 * A296524 A296525 A296526

Keyword

nonn

Author

Zhi-Wei Sun, Dec 14 2017

Status

approved