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

24, 48, 96, 120, 168, 192, 240, 264, 312, 336, 384, 408, 456, 480, 528, 552, 600, 624, 672, 696, 744, 768, 816, 840, 888, 912, 960, 984, 1032, 1056, 1104, 1128, 1176, 1200, 1248, 1272, 1320, 1344, 1392, 1416, 1464, 1488, 1536, 1560, 1608, 1632, 1680, 1704

Offset: 1

Author

Labos Elemer, Aug 08 2002

Keywords

• nonn

Examples

			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
		

Crossrefs

Cf. A045763, A073757-A073762.

Cf. A071904.

Programs

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 *)

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