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A123066
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(Number of numbers <= n with an odd number of distinct prime factors) - (number of numbers <= n with an even number of distinct prime factors).
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5
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0, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 1, 0, -1, 0, -1, -2, -3, -4, -5, -4, -3, -2, -3, -4, -3, -4, -3, -2, -3, -4, -3, -2, -3, -2, -3, -4, -5, -6, -5, -4, -5, -4, -5, -4, -3, -4, -5, -6, -7, -6, -5, -6, -7, -8, -9, -10
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OFFSET
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1,3
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COMMENTS
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Analog of A072203 for number of distinct factors. Conjecture that sequence changes sign infinitely often, although the next sign change is probably large.
The signs first change at n = 52 and then change again at n = 7954. - Harvey P. Dale, Jul 04 2012
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LINKS
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FORMULA
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MAPLE
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a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+
`if`(nops(ifactors(n)[2])::odd, 1, -1))
end:
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MATHEMATICA
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dpf[n_] := Module[{df = PrimeNu[n]}, If[OddQ[df], 1, -1]]; Join[{0}, Accumulate[ Array[dpf, 100, 2]]] (* Harvey P. Dale, Jul 04 2012 *)
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PROG
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(Python)
from sympy import primenu
def A123066(n): return 1+sum(1 if primenu(i)&1 else -1 for i in range(1, n+1)) # Chai Wah Wu, Dec 31 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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