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A273294 Least nonnegative integer m such that there are nonnegative integers x,y,z,w for which x^2 + y^2 + z^2 + w^2 = n and x + 3*y + 5*z = m^2. 13
0, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 3, 3, 3, 4, 0, 1, 2, 3, 2, 3, 3, 4, 4, 0, 1, 2, 3, 3, 4, 4, 2, 3, 3, 4, 0, 1, 2, 3, 4, 2, 3, 6, 4, 3, 3, 6, 4, 0, 1, 2, 2, 3, 5, 4, 4, 4, 3, 4, 5, 5, 3, 4, 0, 1, 2, 3, 4, 5, 4, 6, 4, 3, 4, 4, 4, 3, 4, 4, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Clearly, a(n) = 0 if n is a square. Part (i) of the conjecture in A271518 implies that a(n) always exists.

For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

EXAMPLE

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

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

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

a(3812) = 11 since 3812 = 37^2 + 3^2 + 15^2 + 47^2 with 37 + 3*3 + 5*15 = 11^2.

a(3840) = 16 since 3840 = 48^2 + 16^2 + 32^2 + 16^2 with 48 + 3*16 + 5*32 = 16^2.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

Do[m=0; Label[bb]; Do[If[3y+5z<=m^2&&SQ[n-y^2-z^2-(m^2-3y-5z)^2], Print[n, " ", m]; Goto[aa]], {y, 0, Sqrt[n]}, {z, 0, Sqrt[n-y^2]}]; m=m+1; Goto[bb]; Label[aa]; Continue, {n, 0, 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, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278.

Sequence in context: A201079 A241382 A049260 * A053186 A066628 A255120

Adjacent sequences:  A273291 A273292 A273293 * A273295 A273296 A273297

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 19 2016

STATUS

approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)