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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 16 12:38 EDT 2021. Contains 343947 sequences. (Running on oeis4.)