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A318682
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a(n) is the number of odd values minus the number of even values of the integer log of all positive integers up to and including n.
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1
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-1, -2, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 8, 7, 8, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 13, 12, 11, 12, 13, 14, 13, 14, 13, 14, 15, 14, 15, 16, 17, 16, 15, 14, 15, 16, 15, 14, 15, 14, 15, 16, 17, 18, 17, 16, 17, 18
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OFFSET
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1,2
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COMMENTS
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a(n) = Sum_{k=1..n} (-1)^(sopfr(k)+1), with sopfr(n) the sum of the prime factors of n with repetition, also known as the integer log of n.
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LINKS
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FORMULA
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a(n) = a(n-1) + (-1)^(sopfr(n)+1) with a(1) = (-1)^(sopfr(1)+1) = -1.
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EXAMPLE
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a(4) = -1 - 1 + 1 - 1 = -2, since sopfr(1) = 0, sopfr(2) = 2, sopfr(3) = 3, and sopfr(4) = 4.
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MATHEMATICA
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Nest[Append[#, #[[-1]] + (-1)^(1 + Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[Length@ # + 1] ])] &, {-1}, 79] (* Michael De Vlieger, Sep 10 2018 *)
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PROG
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(Python)
from sympy import factorint
....a_n = 0
....for i in range(1, n+1):
........a_n += (-1)**(sum(p*e for p, e in factorint(i).items())+1)
....return a_n
(PARI) sopfr(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]);
a(n) = sum(k=1, n, (-1)^(sopfr(k)+1)); \\ Michel Marcus, Sep 09 2018
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CROSSREFS
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Cf. A001414 (sum of prime divisors of n with repetition, sopfr(n)).
Cf. A036349 (numbers such that sopfr(n) is even).
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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