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Search: id:A072203
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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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+0
3
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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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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. The smallest counterexample known is a(906180359) = -1 (Lehman).
Note that the keyword is "nonn", even though eventually (beyond the range shown) the terms become negative.
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REFERENCES
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C. B. Haselgrove, A disproof of a conjecture of Polya. Mathematika 5 1958 141-145.
R. S. Lehman, On Liouville's function. Math. Comp. 14 1960 311-320.
G. Polya, Mathematics and Plausible Reasoning, S.8.16.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
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a(n) = 1 - A002819(n) - T. D. Noe, Feb 06 2007
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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}]
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CROSSREFS
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Cf. A028488, A002819, A051470.
Sequence in context: A087039 A102096 A137866 this_sequence A124044 A059981 A033676
Adjacent sequences: A072200 A072201 A072202 this_sequence A072204 A072205 A072206
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KEYWORD
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nonn,nice,easy
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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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Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 13 2002
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