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A063658
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The number of integers m in [1..n] for which gcd(m,n) is divisible by a square greater than 1.
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2
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0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 5, 0, 0, 0, 6, 1, 0, 3, 7, 0, 0, 0, 8, 0, 0, 0, 12, 0, 0, 0, 10, 0, 0, 0, 11, 5, 0, 0, 12, 1, 2, 0, 13, 0, 6, 0, 14, 0, 0, 0, 15, 0, 0, 7, 16, 0, 0, 0, 17, 0, 0, 0, 24, 0, 0, 3, 19, 0, 0, 0, 20, 9, 0, 0, 21, 0, 0, 0, 22, 0, 10, 0, 23, 0, 0, 0, 24
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OFFSET
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1,8
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COMMENTS
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Haviland (1944) proved that a(n) is the number of those arithmetic progressions among (m*n + s: m >= 0), s = 0, 1, ..., n-1, which contain a finite number of squarefree numbers. - Petros Hadjicostas, Jul 21 2019
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s - 1)/zeta(s)*(zeta(s) - zeta(s)/zeta(2*s)). - Geoffrey Critzer, Mar 21 2015
a(n) = Sum_{d|n} phi(n/d) * (1 - mu(d)^2). - Daniel Suteu, Jun 27 2018
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EXAMPLE
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For n=12 we find gcd(4,12), gcd(8,12) and gcd(12,12) divisible by 4, so a(12) = 3.
We have a(2) = 0 because each of the two arithmetic progressions (2*m: m >= 0) and (2*m + 1: m >= 0) contains infinitely many squarefree numbers.
We have a(3) = 0 because each of the three arithmetic progressions (3*m: m >= 0), (3*m + 1: m >= 0), and (3*m + 2: m >= 0) contains infinitely many squarefree numbers.
We have a(4) = 1 because, among the four arithmetic progressions (4*m: m >= 0), (4*m + 1: m >= 0), (4*m + 2: m >= 0), and (4*m + 3: m >= 0), only the first one contains a finite number of squarefree numbers (in this case, zero!).
We have a(8) = 2 because only the arithmetic progressions (8*m: m >= 0) and (8*m + 4: m >= 0) contain a finite number of squarefree numbers (in this case, zero!).
(End)
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MATHEMATICA
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f[list_, i_] := list[[i]]; nn = 100; a =Table[EulerPhi[n], {n, 1, nn}]; b =Table[If[Max[FactorInteger[n][[All, 2]]] > 1, 1, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Mar 21 2015 *)
Table[Sum[EulerPhi[n/d]*(1-MoebiusMu[d]^2), {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
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PROG
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(PARI) { for (n=1, 2000, a=0; for (m=2, n, if (!issquarefree(gcd(m, n)), a++)); write("b063658.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 27 2009
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d) * (1 - moebius(d)^2)); \\ Daniel Suteu, Jun 27 2018
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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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More terms from Larry Reeves (larryr(AT)acm.org), Vladeta Jovovic and Dean Hickerson, Jul 26 2001
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STATUS
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approved
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