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 A306239 Number of ways to write n as x^3 + y^3 + pen(z) + pen(w), where x, y, z, w are nonnegative integers with x <= y and z <= w, and pen(k) denotes the pentagonal number k*(3*k-1)/2. 4
 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 1, 2, 3, 2, 1, 3, 3, 2, 3, 3, 3, 2, 4, 4, 2, 2, 3, 4, 2, 4, 5, 4, 3, 2, 5, 3, 2, 3, 4, 4, 1, 2, 3, 3, 3, 4, 6, 3, 3, 3, 4, 3, 3, 4, 4, 2, 3, 4, 5, 5, 4, 4, 2, 2, 5, 8, 7, 4, 4, 5, 3, 5, 6, 7, 2, 3, 5, 3, 5, 2, 5, 5, 4, 4, 3, 6, 5, 4, 6, 3, 2, 4, 8, 5, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 4, 5, 16, 20, 46. Also, any nonnegative integer not equal to 16 can be written as x^6 + y^3 + pen(z) + pen(w) with x, y, z, w nonnegative integers. We have verified a(n) > 0 for all n = 0..5*10^6. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 EXAMPLE a(4) = 1 with 4 = 1^3 + 1^3 + pen(1) + pen(1). a(5) = 1 with 5 = 0^3 + 0^3 + pen(0) + pen(2). a(16) = 1 with 16 = 2^3 + 2^3 + pen(0) + pen(0). a(20) = 1 with 20 = 0^3 + 2^3 + pen(0) + pen(3). a(46) = 1 with 46 = 1^3 + 1^3 + pen(4) + pen(4). MATHEMATICA PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1, 6]==0); tab={}; Do[r=0; Do[If[PenQ[n-x^3-y^3-z(3z-1)/2], r=r+1], {x, 0, (n/2)^(1/3)}, {y, x, (n-x^3)^(1/3)}, {z, 0, (Sqrt[12(n-x^3-y^3)+1]+1)/6}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab] CROSSREFS Cf. A000326, A000578, A306225, A306227. Sequence in context: A217983 A096626 A083279 * A159455 A105734 A076839 Adjacent sequences: A306236 A306237 A306238 * A306240 A306241 A306242 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 31 2019 STATUS approved

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Last modified May 29 08:58 EDT 2023. Contains 363029 sequences. (Running on oeis4.)