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A182394
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Signs of differences of number of divisors function: a(n) = sign(d(n)-d(n-1)), cf. A000005.
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4
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1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -1, 0, 0, 1, -1, 1, 0, 1, -1, 1, -1, 1, 0, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 0, -1, 1
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OFFSET
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2
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COMMENTS
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d(n) (A000005) has offset 1, being an arithmetic function, so this sequence has offset 2.
Erdős proves that a(n) = 1 with natural density 1/2 and a(n) = -1 with natural density 1/2. Heath-Brown proved that a(n) = 0 infinitely often; see A005237 for details. - Charles R Greathouse IV, Oct 20 2013
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LINKS
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FORMULA
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a(n) = 1 if d(n) > d(n - 1) and a(n) = -1 if d(n) < d(n - 1), otherwise a(n) = 0 if d(n) = d(n - 1), where d(n) is the number of divisors of n (A000005).
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EXAMPLE
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The initial values d(1) ... d(20) are
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, ...
and the first differences are
1, 0, 1, -1, 2, -2, 2, -1, 1, -2, 4, -4, 2, 0, 1, -3, 4, -4, 4, ...,
the signs of which are +1, 0, +1, -1, ...
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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