

A073758


Smallest number that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.


5



0, 0, 0, 0, 0, 4, 0, 6, 6, 4, 0, 8, 0, 4, 6, 6, 0, 4, 0, 6, 6, 4, 0, 9, 10, 4, 6, 6, 0, 4, 0, 6, 6, 4, 10, 8, 0, 4, 6, 6, 0, 4, 0, 6, 6, 4, 0, 9, 14, 4, 6, 6, 0, 4, 10, 6, 6, 4, 0, 8, 0, 4, 6, 6, 10, 4, 0, 6, 6, 4, 0, 10, 0, 4, 6, 6, 14, 4, 0, 6, 6, 4, 0, 8, 10, 4, 6, 6, 0, 4, 14, 6, 6, 4, 10, 9, 0, 4, 6
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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.
a(6) = A073759(6), a(8) = A073759(8), a(9) = A073759(9).
(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, n1, if ((gcd(n, k) != 1) && (n % k), return (k)); ); 0; } \\ Michel Marcus, Mar 29 2016


CROSSREFS

Cf. A045763.
Sequence in context: A291540 A075443 A021250 * A133995 A019629 A073759
Adjacent sequences: A073755 A073756 A073757 * A073759 A073760 A073761


KEYWORD

nonn


AUTHOR

Labos Elemer, Aug 08 2002


STATUS

approved



