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A071553 Least x greater than 1 such that x^n == 1 (mod i) for each i=1,2,3,...,n. 0
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; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..32.

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}] (* Jean-Fran├žois Alcover, Jan 13 2012, after Pari for n <= 10 *)

PROG

(PARI) for(n=1, 12, s=2; while(sum(i=1, n, sign((s^n-1)%i))>0, s++); print1(s, ", "))

CROSSREFS

Sequence in context: A069587 A059843 A092927 * A021812 A155891 A234026

Adjacent sequences:  A071550 A071551 A071552 * A071554 A071555 A071556

KEYWORD

nonn,nice

AUTHOR

Benoit Cloitre, May 30 2002

EXTENSIONS

Edited by Robert G. Wilson v, Jun 07 2002

More terms from Don Reble (djr(AT)nk.ca), Jun 07 2002

Corrected and extended by Vladeta Jovovic, Jun 09 2002

Corrected and extended by Dean Hickerson, Jun 13 2002

STATUS

approved

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Last modified June 28 04:43 EDT 2017. Contains 288813 sequences.