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A071553
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Least x greater than 1 such that x^n == 1 (mod i) for each i=1,2,3,...,n.
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0
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2, 3, 7, 5, 61, 11, 421, 13, 121, 71, 27721, 23, 360361, 4159, 841, 307, 12252241, 1121, 232792561, 2393, 4398241, 483209, 5354228881, 4093, 1460244241, 11232649, 61934401, 7598557, 2329089562801, 406639, 72201776446801, 6998993
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Let m(n) = A003418(n) = LCM(1,2,...,n). Then a(n) <= m(n)+1, with equality if and only if n=1 or n is prime. - David W. Wilson, Vladeta Jovovic, Dean Hickerson.
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MATHEMATICA
| <<NumberTheory`NumberTheoryFunctions` (* Load ChineseRemainder function, needed below. *)
f[n_, m_] := Select[Range[0, m-1], PowerMod[ #, n, m]==1&]; a[1]=2; a[n_] := Module[{lcm, pe, i, m, s, j, x}, lcm=LCM@@Range[n]; pe=Sort[Select[Range[n], Length[FactorInteger[ # ]]==1&&#*FactorInteger[ # ][[1, 1]]>n&], Length[f[n, #1]]/#1<Length[f[n, #2]]/#2&]; For[i=1; m=1; s={0}, i<=Length[pe], i++, s=Union@@Outer[ChineseRemainder[{#1, #2}, {m, pe[[i]]}]&, s, f[n, pe[[i]]]]; m*=pe[[i]]; For[j=2, j<=Length[s], j++, If[PowerMod[x=s[[j]], n, lcm]==1, Return[x]]]; If[PowerMod[1+m, n, lcm]==1, Return[1+m]]; ]]; (* f[n, m] is list of x with x^n==1 (mod m), 0 <= x < m *)
a[1] = 2; a[n_ /; n <= 10] := (s = 2; While[ Sum[ Sign[ Mod[s^n - 1, i]], {i, 1, n}] > 0, s++]; s); a[n_?PrimeQ] := LCM @@ Range[n] + 1; a[n_] := a[n] = (km = If[n <= 24, 6, 7]; redu = Reduce[ And @@ Table[ Mod[x^n, n - k] == 1, {k, 0, km}], x, Integers]; candidates = Join @@ Table[ Sort[ List @@ (redu /. C[1] -> c)[[All, 2]]], {c, 0, n}]; First[ Select[ candidates, # > 1 && And @@ Table[ Mod[ #^n, k] == 1, {k, 2, n - km - 1}] & ]]); Table[ Print[a[n]]; a[n], {n, 1, 32}] (* From Jean-François Alcover, Jan 13 2012, after Pari for n <= 10 *)
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PROG
| (PARI) for(n=1, 12, s=2; while(sum(i=1, n, sign((s^n-1)%i))>0, s++); print1(s, ", "))
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CROSSREFS
| Sequence in context: A069587 A059843 A092927 * A021812 A155891 A069772
Adjacent sequences: A071550 A071551 A071552 * A071554 A071555 A071556
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KEYWORD
| nonn,nice
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), May 30 2002
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EXTENSIONS
| Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 07 2002
More terms from Don Reble (djr(AT)nk.ca), Jun 07 2002
Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 09 2002
Corrected and extended by Dean Hickerson (dean.hickerson(AT)yahoo.com), Jun 13 2002
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