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A238097
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Number of monic cubic polynomials with coefficients from {1..n} and maximum coefficient equal to n, for which all three roots are integers.
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4
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0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 0, 2, 2, 2, 0, 3, 2, 1, 2, 3, 1, 3, 0, 3, 3, 1, 1, 4, 3, 1, 1, 3, 2, 3, 1, 2, 3, 2, 0, 4, 5, 2, 2, 2, 1, 3, 3, 3, 3, 1, 0, 5, 4, 1, 2, 4, 4, 3, 1, 2, 2, 3, 1, 5, 6, 1, 2, 3, 2, 3, 1, 4, 6, 2, 0, 5, 5, 1, 1, 3
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OFFSET
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1,11
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LINKS
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EXAMPLE
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a(11) = 2 with polynomials x^3 + 6*x^2 + 11*x + 6 = (x+1) * (x+2) * (x+3) and x^3 + 7*x^2 + 11*x + 5 = (x+1)^2 * (x+5). - Michael Somos, Feb 23 2014
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MATHEMATICA
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Table[p = Flatten[Table[{a, b, c, 1}, {a, n}, {b, n}, {c, n}], 2]; cnt = 0; Do[If[Max[p[[i]]] == n, poly = p[[i]].x^Range[0, 3]; r = Rest[FactorList[poly]]; If[Total[Transpose[r][[2]]] == 3 && Union[Coefficient[Transpose[r][[1]], x]] == {1}, Print[{n, r}]; cnt++]], {i, Length[p]}]; cnt, {n, 20}] (* T. D. Noe, Feb 22 2014 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sum(a1=1, n, sum(a2=1, n, sum(a3=1, n, vecmax([a1, a2, a3]) == n && vecsum( factor( Pol([1, a1, a2, a3]))[, 2]) == 3))))}; /* Michael Somos, Feb 23 2014 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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