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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
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 famous 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.

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.

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)

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

Cf. A008836, A002819.

Sequence in context: A002819 A307672 A037834 * A179765 A004074 A245615

Adjacent sequences:  A212493 A212494 A212495 * A212497 A212498 A212499

KEYWORD

sign,nice

AUTHOR

Zhi-Wei Sun, May 19 2012

STATUS

approved

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Last modified September 19 15:43 EDT 2020. Contains 337178 sequences. (Running on oeis4.)