A207189 Numbers matching polynomials y(k,x) that have x-1 as a factor; see Comments.
5, 6, 12, 13, 15, 27, 28, 30, 34, 58, 59, 61, 65, 73, 121, 122, 124, 128, 136, 152, 248, 249, 251, 255, 263, 279, 311, 503, 504, 506, 510, 518, 534, 566, 630, 1014, 1015, 1017, 1021, 1029, 1045, 1077, 1141, 1269, 2037, 2038, 2040, 2044, 2052, 2068
Offset: 1
Keywords
Examples
The first 13 polynomials: 1 .... 1 2 .... x 3 .... x + 1 4 .... x^2 5 .... x^2 - 1 6 .... x^2 - x 7 .... x^2 - x - 1 8 .... x^2 + 1 9 .... x^2 + x 10 ... x^2 + x + 1 11 ... x^3 12 ... x^3 - 1 13 ... x^3 - x The list exemplifies these sequences: A207187 (multiples of x + 1): 3,5,9,13,... A207188 (multiples of x): 2,4,6,9,11,13,... A207189 (multiples of x - 1): 5,6,12,13,... A207190 (multiples of x^2 + 1): 8,20,25,27,...
Crossrefs
Cf. A206821.
Programs
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Mathematica
t = Table[IntegerDigits[n, 2], {n, 1, 2000}]; b[n_] := Reverse[Table[x^k, {k, 0, n}]] p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]] TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]] f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1]; q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]] q2[n_] := p[n - f[k] + 2] y1 = Table[p[n], {n, 1, 4}]; Do[AppendTo[y1, Join[Table[q1[n], {n, f[k], g[k] - 1}], Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}] y = Flatten[y1]; (* polynomials over {-1,0,1} *) TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]] Table[y[[n]] /. x -> -1, {n, 1, 300}]; Flatten[Position[%, 0]] (* A207187 *) Table[y[[n]] /. x -> 0, {n, 1, 300}] ; Flatten[Position[%, 0]] (* A207188 *) Table[y[[n]] /. x -> 1, {n, 1, 1200}] ; Flatten[Position[%, 0]] (* A207189 *) Table[y[[n]] /. x -> I, {n, 1, 600}] ; Flatten[Position[%, 0]] (* A207190 *)
Comments