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

%I A167234 #13 Jan 30 2023 19:10:17
%S A167234 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 A167234 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 A167234 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 A167234 Smallest number such that no two divisors of n are congruent modulo a(n).
%C A167234 What can we say about the asymptotic behavior of this sequence? Does it contain every integer > 2 infinitely often?
%C A167234 For n > 6, a(n) <= floor(n/2) + 1; but this seems to be a very crude estimate.
%H A167234 Paul Tek, <a href="/A167234/b167234.txt">Table of n, a(n) for n = 1..10000</a>
%t A167234 allDiffQ[l_List] := (Length[l] == Length[DeleteDuplicates[l]]);
%t A167234 a[n_Integer] := Module[{ds = Divisors[n]},
%t A167234    Catch[Do[If[allDiffQ[Mod[#, m] & /@ ds], Throw[m]], {m, n}]]];
%t A167234 a /@ Range[92] (* _Peter Illig_, Jul 11 2018 *)
%o A167234 (PARI) alldiff(v)=v=vecsort(v);for(k=1,#v-1,if(v[k]==v[k+1],return(0)));1
%o A167234 a(n)=local(ds);ds=divisors(n);for(k=#ds,n,if(alldiff(vector(#ds,i,ds[i]%k)),return(k)))
%o A167234 (Python)
%o A167234 from sympy import divisors
%o A167234 from itertools import count
%o A167234 def a(n):
%o A167234     d = divisors(n)
%o A167234     return next(k for k in count(1) if len(set(di%k for di in d))==len(d))
%o A167234 print([a(n) for n in range(1, 93)]) # _Michael S. Branicky_, Jan 30 2023
%K A167234 nonn
%O A167234 1,2
%A A167234 _Franklin T. Adams-Watters_, Oct 31 2009