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A336504
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3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].
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4
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1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 26, 27, 30, 32, 36, 39, 40, 42, 44, 45, 48, 52, 54, 56, 60, 63, 64, 66, 72, 78, 80, 81, 84, 88, 90, 96, 99, 100, 104, 105, 108, 112, 117, 120, 126, 128, 130, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180
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OFFSET
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1,2
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COMMENTS
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For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 3-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007734(d) * n_d, where A007734(d) is the multiplicative order of 3 modulo the largest divisor of d not divisible by 3, and 0 <= n_d <= phi(d)/A007734(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 41, 258, 1881, 15069, 127350, 1080749, ...
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LINKS
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MATHEMATICA
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rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] > 0]; Select[Range[200], ppQ[#, 3] &]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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