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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 (list; graph; refs; listen; history; text; internal format)
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. 141-145.

R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Kyle Sturgill-Simon, 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, 187-189, 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 !! (n-1)

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: A102096 A137866 A266161 * A217581 A124044 A059981

Adjacent sequences:  A072200 A072201 A072202 * A072204 A072205 A072206

KEYWORD

sign,nice,easy,look,changed

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

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Last modified March 28 02:00 EDT 2017. Contains 284182 sequences.