OFFSET
1,1
FORMULA
Solutions to Mod[A073758(x), 2]=1.
Conjecture: a(n) = 36*n - 18 - 6*(-1)^n = 24 * A001651(n). - Ralf Stephan, Oct 19 2013
The conjecture is false, first counterexample being a(1541) = 55440. - Robert Israel, Sep 11 2014
EXAMPLE
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.
Least odd unrelated number to 55440 is 25 and smallest unrelated (i.e. neither divisor, nor in RRS) to 3603600 is 27.
Question: can be a smallest odd unrelated number be other than a true power of odd prime?
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
MAPLE
A073758:= proc(n) local k;
for k from 2 to n-2 do
if igcd(k, n) > 1 and n mod k > 1 then return k fi
od;
0
end proc:
select(t -> A073758(t)::odd, [$1..1000]); # Robert Israel, Sep 11 2014
MATHEMATICA
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}]
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 08 2002
STATUS
approved