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A340227
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Number of pairs of divisors of n, (d1,d2), such that d1 < d2 and d1*d2 is squarefree.
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0
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0, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 4, 1, 1, 4, 1, 4, 4, 4, 1, 4, 1, 4, 1, 4, 1, 13, 1, 1, 4, 4, 4, 4, 1, 4, 4, 4, 1, 13, 1, 4, 4, 4, 1, 4, 1, 4, 4, 4, 1, 4, 4, 4, 4, 4, 1, 13, 1, 4, 4, 1, 4, 13, 1, 4, 4, 13, 1, 4, 1, 4, 4, 4, 4, 13, 1, 4, 1, 4, 1, 13, 4, 4, 4, 4, 1, 13
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OFFSET
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1,6
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COMMENTS
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If n = p where p is prime, the only pair of divisors of n such that d1 < d2 is (1,p). Since the product 1*p = p is squarefree, this satisfies the constraints. Thus, a(p) = 1 for all p. - Wesley Ivan Hurt, May 21 2021
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LINKS
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FORMULA
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Sum_{d1|n, d2|n, d1<d2} mu(d1*d2)^2, where mu is the Möbius function (A008683).
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EXAMPLE
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a(28) = 4; (1,2), (1,7), (1,14), (2,7)
a(29) = 1; (1,29)
a(30) = 13; (1,2), (1,3), (1,5), (1,6), (1,10), (1,15), (1,30), (2,3), (2,5), (2,15), (3,5), (3,10), (5,6)
a(31) = 1; (1,31)
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MATHEMATICA
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Table[Sum[Sum[MoebiusMu[i*k]^2 (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}]
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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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