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A228409
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a(n) = 4*mu(n) + 5, where mu is the Moebius function (A008683).
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2
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9, 1, 1, 5, 1, 9, 1, 5, 5, 9, 1, 5, 1, 9, 9, 5, 1, 5, 1, 5, 9, 9, 1, 5, 5, 9, 5, 5, 1, 1, 1, 5, 9, 9, 9, 5, 1, 9, 9, 5, 1, 1, 1, 5, 5, 9, 1, 5, 5, 5, 9, 5, 1, 5, 9, 5, 9, 9, 1, 5, 1, 9, 5, 5, 9, 1, 1, 5, 9, 1, 1, 5, 1, 9, 5, 5, 9, 1, 1, 5, 5, 9, 1, 5, 9, 9, 9
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OFFSET
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1,1
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COMMENTS
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If n is prime (A000040), then a(n) = 1. The converse is not true: when n is the product of an odd number of distinct primes, mu(n) = -1 => a(n) = 1 (30 = 2*3*5, so a(30) = 1).
If n is semiprime (A001358), a(n) gives the number of divisors of n^2. In particular, if n = p^2 then n^2 = (p^2)^2 = p^4 has 5 divisors: p^4, p^3, p^2, p, 1. If n = pq (p,q distinct primes) then n^2 = (pq)^2 has 9 divisors: (pq)^2, qp^2, pq^2, p^2, q^2, pq, p, q, and 1.
a(n) = 1 if and only if n has an odd number of distinct prime factors, A030059. - Jon Perry, Nov 12 2013.
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LINKS
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EXAMPLE
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a(6) = 9; 4*mu(6) + 5 = 4*1 + 5 = 9.
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MAPLE
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with(numtheory); A228409:=n->4*mobius(n)+5; seq(A228409(n), n=1..100);
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MATHEMATICA
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Table[4 MoebiusMu[n] + 5, {n, 100}]
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PROG
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CROSSREFS
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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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