A072203 - OEIS

%I #38 Dec 21 2022 04:49:26

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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).

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

%H T. D. Noe, <a href="/A072203/b072203.txt">Table of n, a(n) for n = 1..10000</a>

%H C. B. Haselgrove, <a href="http://dx.doi.org/10.1112/S0025579300001480">A disproof of a conjecture of Polya</a>, Mathematika 5 (1958), pp. 141-145.

%H R. S. Lehman, <a href="http://dx.doi.org/10.1090/S0025-5718-1960-0120198-5">On Liouville's function</a>, Math. Comp., 14 (1960), 311-320.

%H Kyle Sturgill-Simon, <a href="http://www.carroll.edu/library/thesisArchive/Sturgill-Simon_2012final.pdf">An interesting opportunity: the Gilbreath conjecture</a>, Honors Thesis, Mathematics Dept., Carroll College, 2012.

%H M. Tanaka, <a href="http://dx.doi.org/10.3836/tjm/1270216093">A Numerical Investigation on Cumulative Sum of the Liouville Function</a>, Tokyo J. Math. 3:1, 187-189, 1980.

%F a(n) = 1 - A002819(n). - _T. D. Noe_, Feb 06 2007

%t 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}]

%t Join[{0},Accumulate[Rest[Table[If[OddQ[PrimeOmega[n]],1,-1],{n,110}]]]] (* _Harvey P. Dale_, Mar 10 2013 *)

%t Table[1 - Sum[(-1)^PrimeOmega[i], {i, 1, n}], {n, 1, 100}] (* _Indranil Ghosh_, Mar 17 2017 *)

%o (Haskell)

%o a072203 n = a072203_list !! (n-1)

%o a072203_list = scanl1 (\x y -> x + 2*y - 1) a066829_list

%o -- _Reinhard Zumkeller_, Nov 19

%o (PARI) a(n) = 1 - sum(i=1, n, (-1)^bigomega(i));

%o for(n=1, 100, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 17 2017

%o (Python)

%o from functools import reduce

%o from operator import ixor

%o from sympy import factorint

%o 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

%Y Cf. A028488, A002819, A051470, A066829.

%K sign,nice,easy,look

%O 1,3

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

%E Edited and extended by _Robert G. Wilson v_, Jul 13 2002

%E Comment corrected by _Charles R Greathouse IV_, Mar 08 2010