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A066575
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LCM of numbers m such that 1 <= m <= n, m has a common factor with n, but m does not divide n.
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1
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1, 1, 1, 1, 1, 4, 1, 6, 6, 24, 1, 360, 1, 120, 180, 420, 1, 1680, 1, 5040, 1260, 5040, 1, 55440, 60, 55440, 2520, 720720, 1, 10810800, 1, 360360, 83160, 1441440, 2100, 73513440, 1, 24504480, 1081080, 2327925600, 1, 1396755360, 1, 465585120, 5405400, 465585120, 1, 32125373280, 420, 10708457760, 36756720, 53542288800, 1
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OFFSET
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1,6
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COMMENTS
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a(p) = 1 and a(4) = 1, since all 1 <= m <= 4 either divide or are coprime to 4 and 4 is the smallest composite.
a(n) > 1 for composite n > 4.
There are 2 species of m. The first is m | n^e with e >= 2, the second is m that is the product of at least one prime p | n (A272618) and one prime q coprime to n (A272619). Both species of m are composite. We can simply look for composite m in the cototient of n such that m does not divide n.
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LINKS
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FORMULA
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EXAMPLE
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a(12) = 360 = lcm(8,9,10) as 8, 9 and 10 are the only numbers <= 12 which are not relatively prime to 12 nor do they divide 12.
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MAPLE
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for i from 1 to 100 do a := 1:for j from 1 to i do b := igcd(i, j); if(b>1 and b<j) then a := ilcm(a, j):end if:end do:c[i] := a; end do:q := seq(c[k], k=1..100);
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MATHEMATICA
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Table[Apply[LCM, Select[Range[2, n - 1], Nor[Divisible[n, #], CoprimeQ[n, #]] &] /. {} -> 1], {n, 53}] (* Michael De Vlieger, Oct 30 2017 *)
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PROG
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(PARI) A066575(n) = { my(m=1); for(k=1, n, if((n%k)&&(gcd(n, k)>1), m = lcm(m, k))); m; }; \\ Antti Karttunen, Oct 30 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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