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 A299537 Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that x or y is a power of 4 (including 4^0 = 1) and x + 3*y is also a power of 4. 31
 1, 1, 1, 1, 1, 4, 1, 1, 4, 3, 1, 1, 2, 6, 1, 1, 2, 3, 1, 1, 8, 6, 2, 4, 3, 8, 3, 1, 6, 8, 4, 1, 6, 10, 3, 4, 2, 5, 6, 3, 4, 8, 1, 1, 7, 5, 1, 1, 5, 6, 4, 2, 4, 13, 5, 6, 7, 5, 5, 1, 3, 7, 2, 1, 3, 12, 6, 2, 11, 5, 5, 3, 7, 11, 2, 1, 6, 13, 5, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture (i): a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m with k = 0,1,2,... and m = 1, 2, 3, 5, 7, 11, 15, 19, 43, 47, 135, 1103. Conjecture (ii): For any integer n > 1, we can write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 2*x or 2*y is a power of 4 and 2*(x+3*y) is also a power of 4. Note that 81503^2 cannot be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and both x and x + 3*y in the set {4^k: k = 0,1,2,...}. However, 81503^2 = 16372^2 + 4^2 + 52372^2 + 60265^2 with 4 = 4^1 and 16372 + 3*4 = 4^7. We have verified that the conjecture for n up to 10^7. See also the related comments in A300219 and A300360, and a similar conjecture in A299794. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..2000 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. EXAMPLE a(2) = 1 since 2^2 = 1^2 + 1^2 + 1^2 + 1^2 with 1 = 4^0 and 1 + 3*1 = 4^1. a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4 = 4^1 and 4 + 3*0 = 4^1. a(19) = 1 since 19^2 = 1^2 + 0^2 + 6^2 + 18^2 with 1 = 4^0 and 1 + 3*0 = 4^0. a(43) = 1 since 43^2 = 4^2 + 20^2 + 8^2 + 37^2 with 4 = 4^1 and 4 + 3*20 = 4^3. a(135) = 1 since 135^2 = 16^2 + 16^2 + 17^2 + 132^2 with 16 = 4^2 and 16 + 3*16 = 4^3. a(1103) = 1 since 1103^2 = 4^2 + 4^2 + 716^2 + 839^2 with 4 = 4^1 and 4 + 3*4 = 4^2. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; Pow[n_]:=Pow[n]=IntegerQ[Log[4, n]]; tab={}; Do[r=0; Do[If[(Pow[y]||Pow[4^k-3y])&&SQ[n^2-y^2-(4^k-3y)^2-z^2], r=r+1], {k, 0, Log[4, Sqrt*n]}, {y, 0, Min[n, 4^k/3]}, {z, 0, Sqrt[Max[0, (n^2-y^2-(4^k-3y)^2)/2]]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab] CROSSREFS Cf. A000118, A000290, A000302, A271518, A279612, A281976, A299794, A299924, A300219, A300360, A300362. Sequence in context: A070085 A131776 A010322 * A053239 A046569 A046596 Adjacent sequences:  A299534 A299535 A299536 * A299538 A299539 A299540 KEYWORD nonn AUTHOR Zhi-Wei Sun, Mar 04 2018 STATUS approved

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Last modified May 26 14:39 EDT 2020. Contains 334626 sequences. (Running on oeis4.)