A072203 (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).

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

Offset: 1

Author

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

Keywords

• sign
• nice
• easy
• look

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.

Crossrefs

Cf. A028488, A002819, A051470, A066829.

Programs

Haskell

a072203 n = a072203_list !! (n-1)
a072203_list = scanl1 (\x y -> x + 2*y - 1) a066829_list
-- Reinhard Zumkeller, Nov 19 2011

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 *)

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

Formula

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

Extensions

Edited and extended by Robert G. Wilson v, Jul 13 2002

Comment corrected by Charles R Greathouse IV, Mar 08 2010