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 A350857 Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square. 7
 1, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 1, 1, 3, 3, 5, 4, 5, 3, 2, 2, 1, 3, 5, 4, 4, 3, 1, 2, 4, 4, 6, 4, 5, 5, 4, 2, 3, 6, 5, 5, 2, 3, 2, 2, 3, 3, 7, 4, 7, 6, 3, 3, 2, 3, 5, 3, 1, 4, 2, 2, 3, 5, 6, 5, 7, 4, 3, 2, 2, 4, 5, 3, 3, 3, 1, 1, 2, 6, 8, 9, 5, 7, 5, 3, 4, 3, 6, 6, 4, 3, 2, 0, 3, 6, 7, 5, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture 1: a(n) > 0 except for n = 95, 255. This has been verified for n up to 10^7. It seems that a(n) = 1 only for n = 0, 14, 15, 24, 30, 60, 78, 79, 111, 159, 174, 249, 270, 303, 318, 334, 382, 399, 414, 462, 472, 831, 894, 975, 1080, 1151, 1164, 1919, 4080, 6456, 20319. Conjecture 2: For any positive integer a == 2 (mod 4), all sufficiently large integers can be written as a*w^4 + x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If N(a) denotes the largest integer not of the form a*w^4 + x^4 + (2*y)^2 + z^2 (w,x,y,z = 0,1,2,...), then we have N(2) = 255, N(6) = 2716, N(10) = 598, N(14) = 8427, N(18) = 2463, N(22) = 3884, N(26) = 14988, N(30) = 10843. See also A346643 and A347865 for similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120. Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022. EXAMPLE a(24) = 1 with 24 = 2*0^4 + 2^4 + 2^2 + 2^2 and 0 = 0^2. a(1151) = 1 with 1151 = 2*3^4 + 4^4 + 2^2 + 27^2 and 4 = 2^2. a(6456) = 1 with 6456 = 2*1^4 + 3^4 + 17^2 + 78^2 and 1 = 1^2. a(20319) = 1 with 20319 = 2*5^4 + 0^4 + 5^2 + 138^2 and 0 = 0^2. MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; tab={}; Do[r=0; Do[If[SQ[n-2w^4-x^4-y^2]&&(SQ[w]||SQ[x]), r=r+1], {w, 0, (n/2)^(1/4)}, {x, 0, (n-2w^4)^(1/4)}, {y, 0, Sqrt[(n-2w^4-x^4)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab] CROSSREFS Cf. A000290, A000583, A346643, A347865. Sequence in context: A358617 A008968 A162499 * A135715 A089326 A237367 Adjacent sequences: A350854 A350855 A350856 * A350858 A350859 A350860 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 19 2022 STATUS approved

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Last modified December 6 19:33 EST 2022. Contains 358648 sequences. (Running on oeis4.)