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A073763 Least number of unrelated set belonging to these numbers is odd.

%I A073763 #30 Sep 11 2014 17:34:59
%S A073763 24,48,96,120,168,192,240,264,312,336,384,408,456,480,528,552,600,624,
%T A073763 672,696,744,768,816,840,888,912,960,984,1032,1056,1104,1128,1176,
%U A073763 1200,1248,1272,1320,1344,1392,1416,1464,1488,1536,1560,1608,1632,1680,1704
%N A073763 Least number of unrelated set belonging to these numbers is odd.
%F A073763 Solutions to Mod[A073758(x), 2]=1.
%F A073763 Conjecture: a(n) = 36*n - 18 - 6*(-1)^n = 24 * A001651(n). - _Ralf Stephan_, Oct 19 2013
%F A073763 The conjecture is false, first counterexample being a(1541) = 55440. - _Robert Israel_, Sep 11 2014
%e A073763 n=24: UnrelatedSet[24]={9, 10, 14, 15, 16, 18, 20, 21, 22}, Min=9, so 24 is here. In cases of all solutions (<50000) the odd number was always 9. This is not an accident. Primes are either divisors or primes to n. Thus a term here should be a composite odd number from A071904, whose first entry is 9; so next candidates are 15, 21, 25, 27... While 15 and 21 not [yet] found, prime powers 25 and 27 did arise.
%e A073763 Least odd unrelated number to 55440 is 25 and smallest unrelated (i.e. neither divisor, nor in RRS) to 3603600 is 27.
%e A073763 Question: can be a smallest odd unrelated number be other than a true power of odd prime?
%e A073763 Answer: no.  Proof: Suppose A073758(n) = k is odd and not a prime power.  Let k = g*u where g = gcd(n,k) > 1.  Since k does not divide n, u > 1.  Since 2*g < k is not unrelated to n, it must divide n, so n is even.  Let p be a prime factor of u.  Since 2*p is not unrelated to n, p must divide n.  But then p^d < k is unrelated to n, where p^d is the highest power of p dividing k. - _Robert Israel_, Sep 11 2014
%p A073763 A073758:= proc(n) local k;
%p A073763   for k from 2 to n-2 do
%p A073763     if igcd(k,n) > 1 and n mod k > 1 then return k fi
%p A073763   od;
%p A073763   0
%p A073763 end proc:
%p A073763 select(t -> A073758(t)::odd, [$1..1000]); # _Robert Israel_, Sep 11 2014
%t A073763 tn[x_] := Table[w, {w, 1, x}] di[x_] := Divisors[x] rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]] nd[x_] := Complement[tn[x], di[x]] rs[x_] := Union[rrs[x], di[x]] urs[x_] := Complement[tn[x], rs[x]] Do[s=Min[urs[n]]; If[OddQ[s], Print[{n, s}]], {n, 1, 10000}]
%t A073763 unQ[n_] := OddQ[Min[Complement[r = Range[n - 1], Select[r, Divisible[n, #] || GCD[n, #] == 1 &]]]]; Select[Range[1710], unQ] (* _Jayanta Basu_, Jul 09 2013 *)
%Y A073763 Cf. A045763, A073757-A073762.
%Y A073763 Cf. A071904.
%K A073763 nonn
%O A073763 1,1
%A A073763 _Labos Elemer_, Aug 08 2002