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A334368
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Number of proper divisors of n such that d, n/d and n-d are all squarefree.
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1
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0, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 2, 0, 2, 2, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 2, 0, 2, 1, 0, 2, 2, 2, 1, 0, 2, 2, 0, 0, 4, 1, 2, 2, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 2, 0, 2, 1, 0, 0, 3, 1, 2, 2, 3, 1, 0, 0, 2, 1, 2, 2, 3, 1, 0, 0, 1, 1, 4, 0, 2, 2, 0, 0, 1, 1, 1
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OFFSET
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1,6
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COMMENTS
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a(p^k) = mu(p-1)^2 for k = 1 or 2, and 0 for k > 2.
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LINKS
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FORMULA
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a(n) = Sum_{d|n, d<n} mu(d)^2 * mu(n/d)^2 * mu(n-d)^2, where mu is the Möebius function (A008683).
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EXAMPLE
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a(41) = 0; There are no such divisors of 41 since 1 and 41 are squarefree, but 41 - 1 = 40 is not.
a(42) = 4; The four divisors of 42 that meet all three conditions are 1, 3, 7 and 21.
a(43) = 1; The only divisor of 43 that meets all three conditions is 1.
a(44) = 2; The two divisors of 44 that meet all three conditions are 2 and 22.
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MATHEMATICA
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Table[Sum[MoebiusMu[i]^2 MoebiusMu[n/i]^2 MoebiusMu[n - i]^2 (1 - Ceiling[n/i] + Floor[n/i]), {i, Floor[n/2]}], {n, 100}]
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PROG
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(PARI) a(n) = sumdiv(n, d, issquarefree(d) && issquarefree(n-d) && issquarefree(n/d)); \\ Michel Marcus, Apr 25 2020
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CROSSREFS
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Cf. A007427 (with only d and n/d squarefree).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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