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A072203
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(Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).
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4
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0, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 7, 8, 7, 6, 5, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 9, 8, 9, 10, 11, 10, 9, 10, 9, 8, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 2, 3, 4, 3, 4, 5, 6
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OFFSET
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1,3
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COMMENTS
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A number m is oddly or evenly factored depending on whether m has an odd or even number of prime factors, e.g., 12 = 2*2*3 has 3 factors so is oddly factored.
Polya conjectured that a(n) >= 0 for all n, but this was disproved by Haselgrove. Lehman gave the first explicit counterexample, a(906180359) = -1; the first counterexample is at 906150257 (Tanaka).
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REFERENCES
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G. Polya, Mathematics and Plausible Reasoning, S.8.16.
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LINKS
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FORMULA
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MATHEMATICA
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f[n_Integer] := Length[Flatten[Table[ #[[1]], {#[[2]]}] & /@ FactorInteger[n]]]; g[n_] := g[n] = g[n - 1] + If[ EvenQ[ f[n]], -1, 1]; g[1] = 0; Table[g[n], {n, 1, 103}]
Join[{0}, Accumulate[Rest[Table[If[OddQ[PrimeOmega[n]], 1, -1], {n, 110}]]]] (* Harvey P. Dale, Mar 10 2013 *)
Table[1 - Sum[(-1)^PrimeOmega[i], {i, 1, n}], {n, 1, 100}] (* Indranil Ghosh, Mar 17 2017 *)
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PROG
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(Haskell)
a072203 n = a072203_list !! (n-1)
a072203_list = scanl1 (\x y -> x + 2*y - 1) a066829_list
(PARI) a(n) = 1 - sum(i=1, n, (-1)^bigomega(i));
(Python)
from functools import reduce
from operator import ixor
from sympy import factorint
def A072203(n): return 1+sum(1 if reduce(ixor, factorint(i).values(), 0)&1 else -1 for i in range(1, n+1)) # Chai Wah Wu, Dec 20 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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Bill Dubuque (wgd(AT)zurich.ai.mit.edu), Jul 03 2002
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EXTENSIONS
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STATUS
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approved
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