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A349673
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a(n) is the smallest positive integer m such that the set of numbers {f(k) : 1 <= k <= n} are pairwise distinct modulo m for f(x)=x^3+x.
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1
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1, 3, 3, 9, 9, 9, 9, 9, 9, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81, 81
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OFFSET
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1,2
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COMMENTS
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This is called the discriminator of f(x)=x^3+x.
First differs from A064235 at n=244.
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LINKS
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FORMULA
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a(n) = 7*3^(6s+4) if n=3^(6s+5)+1 or n=3^(6s+5)+2; otherwise a(n) = 3^ceiling(log_3(n)) (see Theorem 1.1 in Yang & Zhao).
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MATHEMATICA
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a[n_] := Module[{e1 = IntegerExponent[n-1, 3], e2 = IntegerExponent[n-2, 3], e = -1}, If[3^e1 == n - 1 && Mod[e1, 6] == 5, e = e1 - 1]; If[3^e2 == n-2 && Mod[e2, 6] == 5, e = e2 - 1]; If[e > -1, 7*3^e, 3^Ceiling[Log[3, n]]]]; Array[a, 100] (* Amiram Eldar, Nov 24 2021 *)
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PROG
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(PARI) isok(n, m) = my(v=vector(n, k, (k^3+k)% m)); #v == #vecsort(v, , 8);
a(n) = my(m=1); while (!isok(n, m), m++); m;
(PARI) a(n) = {my(v); if ((n%3 == 1) && (n!=1), v=valuation(n-1, 3); if ((3^v == n-1) && ((v%6)==5), return(7*3^(v-1)))); if ((n%3 == 2) && (n!=2), v=valuation(n-2, 3); if ((3^v == n-2) && ((v%6)==5), return(7*3^(v-1)))); return(3^ceil(log(n)/log(3))); }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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