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 A262827 Number of ordered ways to write n as w^2 + x^3 + y^4 + 2*z^4, where w, x, y and z are nonnegative integers. 30
 1, 3, 4, 4, 4, 3, 2, 2, 2, 3, 4, 4, 4, 2, 1, 1, 2, 5, 5, 5, 4, 1, 1, 1, 2, 4, 5, 6, 6, 3, 3, 2, 3, 7, 6, 4, 4, 5, 4, 3, 3, 4, 5, 4, 5, 4, 3, 2, 2, 8, 5, 3, 6, 4, 3, 2, 2, 5, 4, 4, 5, 2, 1, 2, 5, 9, 7, 5, 7, 4, 3, 1, 2, 4, 3, 5, 5, 2, 1, 3, 3, 8, 9, 8, 8, 5, 2, 1, 2, 5, 6, 7, 7, 3, 2, 2, 4, 7, 7, 2, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: a(n) > 0 for all n >= 0. Also, a(n) = 1 only for the following 41 values of n: 0, 14, 15, 21, 22, 23, 62, 71, 78, 87, 136, 216, 405, 437, 448, 477, 535, 583, 591, 623, 671, 696, 885, 950, 1046, 1135, 1206, 1208, 1248, 1317, 2288, 2383, 2543, 3167, 3717, 3974, 6847, 7918, 8328, 9096, 21935. We have verified that a(n) > 0 for all n = 0..10^7. We also conjecture that if f(w,x,y,z) is one of the 8 polynomials 2w^2+x^3+4y^3+z^4, w^2+x^3+2y^3+c*z^3 (c = 3,4,5,6) and w^2+x^3+2y^3+d*z^4 (d = 1,3,6) then each n = 0,1,2,... can be written as f(w,x,y,z) with w,x,y,z nonnegative integers. - Zhi-Wei Sun, Dec 30 2017 LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. (See Remark 1.1.) Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. (See Theorem 1.1 and Conjecture 1.1.) EXAMPLE a(14) = 1 since 14 = 2^2 + 2^3 + 0^4 + 2*1^4. a(87) = 1 since 87 = 2^2 + 0^3 + 3^4 + 2*1^4. a(216) = 1 since 216 = 0^2 + 6^3 + 0^4 + 2*0^4. a(405) = 1 since 405 = 18^2 + 0^3 + 3^4 + 2*0^4. a(1248) = 1 since 1248 = 31^2 + 5^3 + 0^4 + 2*3^4. a(1317) = 1 since 1317 = 23^2 + 1^3 + 5^4 + 2*3^4. a(2288) = 1 since 2288 = 44^2 + 4^3 + 4^4 +2*2^4. a(2383) = 1 since 2383 = 1462 + 9^3 + 6^4 + 2*3^4. a(2543) = 1 since 2543 = 50^2 + 3^3 + 2^4 + 2*0^4. a(3167) = 1 since 3167 = 54^2 + 2^3 + 3^4 + 2*3^4. a(3717) = 1 since 3717 = 18^2 + 15^3 + 2^4 + 2*1^4. a(3974) = 1 since 3974 = 39^2 + 13^3 + 4^4 + 2*0^4. a(6847) = 1 since 6847 = 52^2 + 15^3 + 4^4 + 2*4^4. a(7918) = 1 since 7918 = 46^2 + 10^3 + 0^4 + 2*7^4. a(8328) = 1 since 8328 = 42^2 + 1^3 + 9^4 + 2*1^4. a(9096) = 1 since 9096 = 44^2 + 18^3 + 6^4 + 2*2^4. a(21935) = 1 since 21935 = 66^2 + 26^3 + 1^4 + 2*1^4. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] Do[r=0; Do[If[SQ[n-x^3-y^4-2*z^4], r=r+1], {x, 0, n^(1/3)}, {y, 0, (n-x^3)^(1/4)}, {z, 0, ((n-x^3-y^4)/2)^(1/4)}]; Print[n, " ", r]; Continue, {n, 0, 100}] CROSSREFS Cf. A000290, A000578, A000583, A262813, A262815, A262816, A262824. Sequence in context: A014241 A199185 A279781 * A143490 A007485 A280356 Adjacent sequences:  A262824 A262825 A262826 * A262828 A262829 A262830 KEYWORD nonn AUTHOR Zhi-Wei Sun, Oct 03 2015 STATUS approved

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Last modified October 14 16:48 EDT 2019. Contains 328022 sequences. (Running on oeis4.)