

A072203


(Number of oddly factored numbers <= n)  (number of evenly factored numbers <= n).


4



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

1,3


COMMENTS

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).


REFERENCES

G. Polya, Mathematics and Plausible Reasoning, S.8.16.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
C. B. Haselgrove, A disproof of a conjecture of Polya, Mathematika 5 (1958), pp. 141145.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311320.
Kyle SturgillSimon, An interesting opportunity: the Gilbreath conjecture, Honors Thesis, Mathematics Dept., Carroll College, 2012.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3:1, 187189, 1980.


FORMULA

a(n) = 1  A002819(n).  T. D. Noe, Feb 06 2007


MATHEMATICA

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 *)


PROG

(Haskell)
a072203 n = a072203_list !! (n1)
a072203_list = scanl1 (\x y > x + 2*y  1) a066829_list
 Reinhard Zumkeller, Nov 19
(PARI) a(n) = 1  sum(i=1, n, (1)^bigomega(i));
for(n=1, 100, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 17 2017


CROSSREFS

Cf. A028488, A002819, A051470, A066829.
Sequence in context: A137866 A329888 A266161 * A217581 A124044 A059981
Adjacent sequences: A072200 A072201 A072202 * A072204 A072205 A072206


KEYWORD

sign,nice,easy,look


AUTHOR

Bill Dubuque (wgd(AT)zurich.ai.mit.edu), Jul 03 2002


EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 13 2002
Comment corrected by Charles R Greathouse IV, Mar 08 2010


STATUS

approved



