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A073759
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Largest number that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.
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6
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0, 0, 0, 0, 0, 4, 0, 6, 6, 8, 0, 10, 0, 12, 12, 14, 0, 16, 0, 18, 18, 20, 0, 22, 20, 24, 24, 26, 0, 28, 0, 30, 30, 32, 30, 34, 0, 36, 36, 38, 0, 40, 0, 42, 42, 44, 0, 46, 42, 48, 48, 50, 0, 52, 50, 54, 54, 56, 0, 58, 0, 60, 60, 62, 60, 64, 0, 66, 66, 68, 0, 70, 0, 72, 72, 74, 70, 76
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OFFSET
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1,6
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COMMENTS
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Largest "unrelated" number to n.
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)
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LINKS
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EXAMPLE
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n = 20: unrelated set to 20 = {6,8,12,14,15,16,18},largest is a(20) = 18.
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MATHEMATICA
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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[Max[unr[w]], {w, 1, 128}]; (for empty sets 0 was used).
Table[t = Select[r = Range[n - 1], Divisible[n, #] || GCD[n, #] == 1 &]; Max[Join[{0}, Complement[r, t]]], {n, 78}] (* Jayanta Basu, Jul 09 2013 *)
Table[SelectFirst[Range[n - 2, 2, -1], 1 < GCD[#, n] < # &] /. n_ /; MissingQ@ n -> 0, {n, 100}] (* Michael De Vlieger, Mar 28 2016, Version 10.2 *)
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PROG
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(PARI) a(n) = {forstep(k=n-2, 1, -1, if ((gcd(n, k) != 1) && (n % k), return (k)); ); 0; } \\ Michel Marcus, Mar 29 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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