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A326440
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a(n) = 1 - tau(1) + tau(2) - tau(3) + ... + (-1)^n tau(n), where tau = A000005 is number of divisors.
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1
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1, 0, 2, 0, 3, 1, 5, 3, 7, 4, 8, 6, 12, 10, 14, 10, 15, 13, 19, 17, 23, 19, 23, 21, 29, 26, 30, 26, 32, 30, 38, 36, 42, 38, 42, 38, 47, 45, 49, 45, 53, 51, 59, 57, 63, 57, 61, 59, 69, 66, 72, 68, 74, 72, 80, 76, 84, 80, 84, 82, 94, 92, 96, 90, 97, 93, 101, 99
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OFFSET
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0,3
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COMMENTS
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Is this sequence nonnegative?
As tau(n) is odd when n is a square, there are alternating strings of even and odd integers with change of parity for each n square. Indeed, between m^2 and (m+1)^2-1, there is a string of 2m+1 even terms if m is odd, or a string of 2m+1 odd terms if m is even. - Bernard Schott, Jul 10 2019
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LINKS
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Michel Marcus, Table of n, a(n) for n = 0..5000
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FORMULA
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a(n) = 1 + Sum_{k=1..n} (-1)^k A000005(k).
For n > 0, a(n) = 1 + A307704(n).
If p prime, a(p) = a(p-1) - 2. - Bernard Schott, Jul 10 2019
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EXAMPLE
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The first 6 terms of A000005 are 1, 2, 2, 3, 2, 4, so a(6) = 1 - 1 + 2 - 2 + 3 - 2 + 4 = 5.
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MATHEMATICA
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Accumulate[Table[If[k==0, 1, (-1)^k*DivisorSigma[0, k]], {k, 0, 30}]]
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PROG
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(PARI) a(n) = 1 - sum(k=1, n, (-1)^(k+1)*numdiv(k)); \\ Michel Marcus, Jul 09 2019
(MAGMA) [1] cat [1+(&+[(-1)^(k)*#Divisors(k):k in [1..n]]):n in [1..70]]; // Marius A. Burtea, Jul 10 2019
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CROSSREFS
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Cf. A000005, A001222, A008683, A054519, A071321, A195017, A268387, A307704, A316523, A316524, A319273.
Sequence in context: A282892 A008798 A005290 * A166117 A078051 A130627
Adjacent sequences: A326437 A326438 A326439 * A326441 A326442 A326443
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Jul 06 2019
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STATUS
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approved
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