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A106309
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Primes p such that for all initial conditions (x(0),x(1),x(2),x(3),x(4)) in [0..p-1]^5 except [0,0,0,0,0], the 5-step recurrence x(k) = x(k-1) + x(k-2) + x(k-3) + x(k-4) + x(k-5) (mod p) has the same period.
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6
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5, 7, 11, 13, 17, 31, 37, 41, 53, 79, 107, 199, 233, 239, 311, 331, 337, 389, 463, 523, 541, 547, 557, 563, 577, 677, 769, 853, 937, 971, 1009, 1021, 1033, 1049, 1061, 1201, 1237, 1291, 1307, 1361, 1427, 1453, 1543, 1657, 1699, 1723, 1747, 1753, 1759, 1787, 1801, 1811, 1861, 1877, 1997, 1999
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OFFSET
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1,1
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COMMENTS
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The first term not in A371566 is a(105) = 4259.
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LINKS
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EXAMPLE
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a(3) = 11 is a term because the recurrence has period 16105 for all initial conditions except (0,0,0,0,0).
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MAPLE
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filter:= proc(p) local Q, q, F, i, z, d, k, kp, G, alpha;
Q:= z^5 - z^4 - z^3 - z^2 - z - 1;
if Irreduc(Q) mod p then return true fi;
F:= (Factors(Q) mod p)[2];
if ormap(t -> t[2]>1, F) then return false fi;
for i from 1 to nops(F) do
q:= F[i][1];
d:= degree(q);
if d = 1 then
kp:= numtheory:-order(solve(q, z), p);
else
G:= GF(p, d, q);
alpha:= G:-ConvertIn(z);
kp:= G:-order(alpha);
fi;
if i = 1 then k:= kp
elif kp <> k then return false
fi;
od;
true
end proc:
select(filter, [seq(ithprime(i), i=1..1000)]);
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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4259 found by D. S. McNeil.
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STATUS
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approved
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