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 A272888 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w*(x^2 + 8*y^2 - z^2) a square, where w,x,y are nonnegative integers and z is a positive integer. 19
 1, 2, 2, 1, 4, 5, 1, 2, 5, 5, 4, 4, 5, 8, 2, 2, 8, 6, 4, 6, 9, 5, 3, 4, 5, 12, 9, 1, 11, 8, 4, 2, 8, 9, 8, 7, 6, 12, 1, 5, 14, 10, 4, 8, 15, 9, 3, 4, 8, 14, 11, 5, 11, 16, 2, 6, 11, 6, 11, 4, 13, 13, 1, 1, 16, 17, 6, 9, 13, 9, 5, 7, 9, 19, 12, 6, 17, 8, 4, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 7, 39, 63, 87, 5116, 2^(4k+2)*m (k = 0,1,2,... and m = 1, 7). See arXiv:1604.06723 for more refinements of Lagrange's four-square theorem. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016. Zhi-Wei Sun, Refine Lagrange's four-square theorem, a message to Number Theory List, April 26, 2016. EXAMPLE a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 0*(0^2 + 8*0^2 - 1^2) = 0^2. a(4) = 1 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 2 > 0 and 0*(0^2 + 8*0^2 - 2^2) = 0^2. a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and 2*(1^2 + 8*1^2 - 1^2) = 4^2. a(28) = 1 since 28 = 2^2 + 2^2 + 4^2 + 2^2 with 2 > 0 and 2*(2^2 + 8*4^2 - 2^2) = 16^2. a(39) = 1 since 39 = 1^2 + 3^2 + 2^2 + 5^2 with 5 > 0 and 1*(3^2 + 8*2^2 - 5^2) = 4^2. a(63) = 1 since 63 = 2^2 + 5^2 + 3^2 + 5^2 with 5 > 0 and 2*(5^2 + 8*3^2 - 5^2) = 12^2. a(87) = 1 since 87 = 2^2 + 1^2 + 9^2 + 1^2 with 1 > 0 and 2*(1^2 + 8*9^2 - 1^2) = 36^2. a(5116) = 1 since 5116 = 65^2 + 9^2 + 9^2 + 27^2 with 27 > 0 and 65*(9^2 + 8*9^2 - 27^2) = 0^2. MAPLE N:= 1000; # to get a(1)..a(N) A:= Vector(N): for z from 1 to floor(sqrt(N)) do   for x from 0 to floor(sqrt(N-z^2)) do     for y from 0 to floor(sqrt(N-z^2-x^2)) do       q:= x^2 + 8*y^2 - z^2;       if q < 0 then         A[x^2+y^2+z^2]:= A[x^2+y^2+z^2]+1       elif q = 0 then         for w from 0 to floor(sqrt(N-z^2-x^2-y^2)) do            m:= w^2 + x^2 + y^2 + z^2;            A[m]:= A[m]+1;         od       else         wm:= mul(`if`(t[2]::odd, t[1], 1), t=isqrfree(q)[2]);         for j from 0 to floor((N-z^2-x^2-y^2)^(1/4)/sqrt(wm)) do            m:= (wm*j^2)^2 + x^2 + y^2 + z^2;            A[m]:= A[m]+1;         od;       fi     od   od od: convert(A, list); # Robert Israel, May 27 2016 MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] Do[r=0; Do[If[SQ[n-x^2-y^2-z^2]&&SQ[Sqrt[n-x^2-y^2-z^2](x^2+8y^2-z^2)], r=r+1], {x, 0, Sqrt[n-1]}, {y, 0, Sqrt[n-1-x^2]}, {z, 1, Sqrt[n-x^2-y^2]}]; Print[n, " ", r]; Continue, {n, 1, 80}] CROSSREFS Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620. Sequence in context: A238870 A213946 A145036 * A001404 A104580 A202193 Adjacent sequences:  A272885 A272886 A272887 * A272889 A272890 A272891 KEYWORD nonn AUTHOR Zhi-Wei Sun, May 08 2016 EXTENSIONS Rick L. Shepherd, May 27 2016: I checked all the statements in each example. STATUS approved

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Last modified January 25 07:50 EST 2020. Contains 331241 sequences. (Running on oeis4.)