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A167234 Smallest number such that no two divisors of n are congruent modulo a(n). 2


%S 1,2,3,4,3,6,4,5,5,6,3,7,5,8,8,9,3,10,4,7,8,6,3,13,7,7,5,11,3,11,4,9,

%T 7,6,8,13,5,5,7,11,3,16,4,12,13,6,3,17,5,11,9,7,3,10,7,15,5,5,3,21,7,

%U 7,11,11,7,14,4,7,7,16,3,13,5,10,13,7,8,14,4,17,7,6,3,23,9,8,5,13,3,19,8,12

%N Smallest number such that no two divisors of n are congruent modulo a(n).

%C What can we say about the asymptotic behavior of this sequence? Does it contain every integer > 2 infinitely often?

%C For n > 6, a(n) <= floor(n/2) + 1; but this seems to be a very crude estimate.

%H Paul Tek, <a href="/A167234/b167234.txt">Table of n, a(n) for n = 1..10000</a>

%t allDiffQ[l_List] := (Length[l] == Length[DeleteDuplicates[l]]);

%t a[n_Integer] := Module[{ds = Divisors[n]},

%t Catch[Do[If[allDiffQ[Mod[#, m] & /@ ds], Throw[m]], {m, n}]]];

%t a /@ Range[92] (* _Peter Illig_, Jul 11 2018 *)

%o (PARI) alldiff(v)=v=vecsort(v);for(k=1,#v-1,if(v[k]==v[k+1],return(0)));1

%o a(n)=local(ds);ds=divisors(n);for(k=#ds,n,if(alldiff(vector(#ds,i,ds[i]%k)),return(k)))

%K nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Oct 31 2009

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Last modified December 6 18:16 EST 2021. Contains 349567 sequences. (Running on oeis4.)