A212496 a(n) = Sum_{k=1..n} (-1)^(k-Omega(k)) with Omega(k) the total number of prime factors of k (counted with multiplicity).

-1, -2, -1, 0, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 5, 6, 5, 6, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 7, 8, 7, 8, 9, 10

Offset: 1

Author

Zhi-Wei Sun, May 19 2012

Keywords

• sign
• nice

Comments

On May 16 2012, Zhi-Wei Sun conjectured that a(n) is positive for each n > 4. He has verified this for n up to 10^10, and shown that the conjecture implies the Riemann Hypothesis. Moreover, he guessed that a(n) > sqrt(n) for any n > 324 (and also a(n) < sqrt(n)*log(log(n)) for n > 5892); this implies that the sequence contains all natural numbers.

Sun also conjectured that b(n) = Sum_{k=1..n} (-1)^(k-Omega(k))/k < 0 for all n=1,2,3,..., and verified this for n up to 2*10^9. Moreover, he guessed that b(n) < -1/sqrt(n) for all n > 1, and b(n) > -log(log(n))/sqrt(n) for n > 2008.

Examples

			We have a(4)=0 since (-1)^(1-Omega(1)) + (-1)^(2-Omega(2)) + (-1)^(3-Omega(3)) + (-1)^(4-Omega(4)) = -1 - 1 + 1 + 1 = 0.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, On a pair of zeta functions, preprint, arxiv:1204.6689 [math.NT], 2012-2016.
• Zhi-Wei Sun, On the parities of Omega(n)-n, a message to Number Theory List, May 18, 2012.
• Zhi-Wei Sun, Table of n, a(n) for n = 1..10^7 (rar-compressed)

Crossrefs

Cf. A008836, A002819.

Programs

Maple

ListTools:-PartialSums([seq((-1)^(k-numtheory:-bigomega(k)),k=1..60)]); # Robert Israel, Jan 03 2023

Mathematica

PrimeDivisor[n_]:=Part[Transpose[FactorInteger[n]],1]
Omega[n_]:=If[n==1,0,Sum[IntegerExponent[n,Part[PrimeDivisor[n],i]],{i,1,Length[PrimeDivisor[n]]}]]
s[0]=0
s[n_]:=s[n]=s[n-1]+(-1)^(n-Omega[n])
Do[Print[n," ",s[n]],{n,1,100000}]
Accumulate[Table[(-1)^(n-PrimeOmega[n]),{n,1000}]] (* Harvey P. Dale, Oct 07 2013 *)

PARI

a(n)=sum(k=1,n, (-1)^(bigomega(k)+k)) \\ Charles R Greathouse IV, Jul 31 2016

Python

from functools import reduce
from operator import ixor
from sympy import factorint
def A212496(n): return sum(-1 if reduce(ixor, factorint(i).values(),i)&1 else 1 for i in range(1,n+1)) # Chai Wah Wu, Jan 03 2023