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 A282163 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^3. 7
 1, 154836, 985320, 1108536, 1113959, 1492260, 1576696, 1632708, 1649238, 1684540, 1805570, 1988008, 2508792, 2548810, 2550408, 2659260, 2698740, 2746590, 2995122, 3074552, 3286710, 3330795, 3538458, 3574200, 3730155, 4039932, 4160240, 4318548, 4374370, 4426695, 4523985 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^3. - Lucian Craciun, Feb 09 2017 The asymptotic density of this sequence is 0.000031511777... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021 LINKS Lucian Craciun, Table of n, a(n) for n = 1..16000 (corrected and extended by Giovanni Resta) Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020. Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient. EXAMPLE The central binomial coefficient C(2*154836,154836) is divisible by 154836^3. MAPLE A282163 := proc (n, m) local a, cbc, k; a := {}; cbc := binomial(2*n, n); for k from n+1 to m do cbc := cbc*(4-2/k); if type(cbc/k^3, integer) then a := `union`(a, {k}) end if end do; a end proc; A282163(0, 10^6) MATHEMATICA Select[Table[n, {n, 10^6}], IntegerQ[Binomial[2#, #]/#^3] &] (* for small n *) n := 0; m := 10^6; A282163 := {}; cbc := Binomial[2n, n]; For[k := n+1, k <= m, k++, {cbc *= 4-2/k, If[IntegerQ[cbc/k^3], A282163 = Append[A282163, k]]}] (* for large m *) A282163:={}; k:=3; For[n:=1, n<=10^6, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]

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Last modified September 9 14:16 EDT 2024. Contains 375764 sequences. (Running on oeis4.)