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.
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
(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
CROSSREFS
KEYWORD
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