OFFSET
1,6
COMMENTS
Original name: Smallest number of "unrelated set" belonging to n [=URS(n)]. Least number, neither divisor nor relatively prime to n. Or a(n)=0 if unrelated set is empty.
From Michael De Vlieger, Mar 28 2016 (Start):
Primes n have no unrelated numbers m < n since all such numbers are coprime to n.
Unrelated numbers m must be composite since primes must either divide or be coprime to n.
m = 1 is not counted as unrelated as it divides and is coprime to n.
a(4) = 0 since 4 is the smallest composite and unrelated numbers m with respect to n must be composite and smaller than n. All other composite n have at least one unrelated number m.
The test for unrelated numbers m that belong to n is 1 < gcd(m, n) < m.
(End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
a(20) = 6 since it is the smallest term of the set of numbers m that neither divide nor are coprime to 20, i.e., {6, 8, 12, 14, 15, 16, 18}.
MATHEMATICA
tn[x_] := Table[w, {w, 1, x}]; di[x_] := Divisors[x]; dr[x_] := Union[di[x], rrs[x]]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; unr[x_] := Complement[tn[x], dr[x]]; Table[Min[unr[w]], {w, 1, 128}]. (* + or -Infinity is replaced by 0 *)
Table[SelectFirst[Range[4, n - 2], 1 < GCD[#, n] < # &] /. n_ /; MissingQ@ n -> 0, {n, 99}] (* Michael De Vlieger, Mar 28 2016, Version 10.2 *)
PROG
(PARI) a(n) = {for(k=1, n-1, if ((gcd(n, k) != 1) && (n % k), return (k)); ); 0; } \\ Michel Marcus, Mar 29 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 08 2002
STATUS
approved