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A206284
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Numbers that match irreducible polynomials over the nonnegative integers.
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28
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3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 28, 30, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 76, 80, 81, 88, 92, 96, 98, 100, 102, 104, 108, 112, 114, 116, 118, 120, 124, 126, 130, 132, 136, 140, 144, 148, 150, 152, 154, 160, 162, 164, 168, 170
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OFFSET
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1,1
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COMMENTS
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Starting with 1, which encodes 0-polynomial, each integer m encodes (or "matches") a polynomial p(m,x) with nonnegative integer coefficients determined by the prime factorization of m. Write m = prime(1)^e(1) * prime(2)^e(2) * ... * prime(k)^e(k); then p(m,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
Identities:
p(m*n,x) = p(m,x) + p(n,x),
p(m*n,x) = p(gcd(m,n),x) + p(lcm(m,n),x),
p(m+n,x) = p(gcd(m,n),x) + p((m+n)/gcd(m,n),x), so that if A003057 is read as a square matrix, then
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LINKS
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EXAMPLE
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Polynomials having nonnegative integer coefficients are matched to the positive integers as follows:
m p(m,x) irreducible
---------------------------
1 0 no
2 1 no
3 x yes
4 2 no
5 x^2 no
6 1+x yes
7 x^3 no
8 3 no
9 2x yes
10 1+x^2 yes
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MAPLE
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P:= n -> add(f[2]*x^(numtheory:-pi(f[1])-1), f = ifactors(n)[2]):
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MATHEMATICA
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b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 400;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x], {n, 1, z/4}]
v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[v, n]], {n, z/2}]; v (* A206284 *)
Complement[Range[200], v] (* A206285 *)
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PROG
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(PARI) is(n)=my(f=factor(n)); polisirreducible(sum(i=1, #f[, 1], f[i, 2]*'x^primepi(f[i, 1]-1))) \\ Charles R Greathouse IV, Feb 12 2012
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CROSSREFS
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Terms of A277318 form a proper subset of this sequence. Cf. also A277316.
Other sequences about factorization in the same polynomial ring: A206442, A284010.
Polynomial multiplication using the same encoding: A297845.
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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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