cp's OEIS Frontend

A066829 Parity of Omega(n): a(n) = 1 if n is the product of an odd number of primes; 0 if product of even number of primes.

%I A066829 #68 Apr 09 2025 22:50:38
%S A066829 0,1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0,1,1,1,1,1,1,0,0,
%T A066829 0,0,1,0,0,0,1,1,1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0,1,0,1,0,1,0,0,1,1,1,
%U A066829 0,1,1,1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1,1,1,0,1,1,1,0
%N A066829 Parity of Omega(n): a(n) = 1 if n is the product of an odd number of primes; 0 if product of even number of primes.
%C A066829 From _Reinhard Zumkeller_, Jul 01 2009: (Start)
%C A066829 The first N Terms are constructed by the following sieving process:
%C A066829 for j:=1 until N do a(j):=0,
%C A066829 for i:=1 until N/2 do
%C A066829 for j:=2*i step i until N do a(j):=1-a(i). (End)
%C A066829 Omega is also written in the OEIS as bigomega. See also comments, references and formulas in A008836 (Liouville's lambda), A007421 and A065043, that all contain the same information as this sequence. - _Antti Karttunen_, Apr 30 2022
%H A066829 Antti Karttunen, <a href="/A066829/b066829.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from Reinhard Zumkeller)
%H A066829 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper4/page1.htm">Irregular numbers</a>, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21 (provides Dirichlet g.f.)
%H A066829 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A066829 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%H A066829 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A066829 a(A026424(n)) = 1; a(A028260(n)) = 0.
%F A066829 Dirichlet g.f.: (zeta(s)^2 - zeta(2*s)) / (2*zeta(s)). [Typo corrected by _Vaclav Kotesovec_, Jan 30 2024]
%F A066829 a(n) = (1-A008836(n)) / 2. - Corrected by _Antti Karttunen_, Apr 30 2022
%F A066829 a(m*n) = a(m) XOR a(n). - _Reinhard Zumkeller_, Aug 28 2008
%F A066829 a(n) = A001222(n) mod 2. - _Reinhard Zumkeller_, Nov 19 2011
%F A066829 From _Antti Karttunen_, May 01 & Nov 30 2022: (Start)
%F A066829 a(n) = 1 - A065043(n) = A349905(n) mod 2.
%F A066829 a(n) = A353556(n) + A353558(n).
%F A066829 a(n) = A358751(n) + A358753(n). (End)
%F A066829 a(n) = A000035(A001222(n)). - _Omar E. Pol_, Apr 09 2025
%e A066829 From _Reinhard Zumkeller_, Jul 01 2009: (Start)
%e A066829 Sieve for N = 30, also demonstrating the affinity to the Sieve of Eratosthenes:
%e A066829 [initial] a(j):=0, 1<=j<=30:
%e A066829 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e A066829 [i=1] a(1)=0 --> a(j):=1, 2<=j<=30:
%e A066829 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A066829 [i=2] a(2)=1 --> a(2*j):=0, 2<=j<=[30/2]:
%e A066829 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
%e A066829 [i=3] a(3)=1 --> a(3*j):=0, 2<=j<=[30/3]:
%e A066829 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0
%e A066829 [i=4] a(4)=0 --> a(4*j):=1, 2<=j<=[30/4]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0
%e A066829 [i=5] a(5)=1 --> a(5*j):=0, 2<=j<=[30/5]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0
%e A066829 [i=6] a(6)=0 --> a(6*j):=1, 2<=j<=[30/6]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1
%e A066829 [i=7] a(7)=1 --> a(7*j):=0, 2<=j<=[30/7]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1
%e A066829 [i=8] a(8)=1 --> a(8*j):=0, 2<=j<=[30/8]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1
%e A066829 [i=9] a(9)=0 --> a(9*j):=1, 2<=j<=[30/9]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1
%e A066829 [i=10] a(10)=0 --> a(10*j):=1, 2<=j<=[30/10]:
%e A066829 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1
%e A066829 and so on: a(22):=0 in [i=11], a(24):=0 in [i=12], a(26):=0 in [i=13], a(28):=1 in [i=14], and a(30):=1 in [i=15]. (End)
%p A066829 A066829 := proc(n)
%p A066829     modp(numtheory[bigomega](n) ,2) ;
%p A066829 end proc:
%p A066829 seq(A066829(n),n=1..80) ; # _R. J. Mathar_, Jul 15 2017
%t A066829 Table[(1-LiouvilleLambda[n])/2,{n,1,20}] (* _Enrique Pérez Herrero_, Jul 07 2012 *)
%t A066829 Table[If[OddQ[PrimeOmega[n]],1,0],{n,120}] (* _Harvey P. Dale_, Mar 12 2016 *)
%o A066829 (PARI) A066829(n) = (bigomega(n)%2); \\ Simplified by _Antti Karttunen_, Apr 30 2022
%o A066829 (Haskell)
%o A066829 a066829 = (`mod` 2) . a001222 -- _Reinhard Zumkeller_, Nov 19 2011
%o A066829 (Python)
%o A066829 from sympy import primeomega as Omega
%o A066829 def a(n): return Omega(n)%2
%o A066829 print([a(n) for n in range(1, 105)]) # _Michael S. Branicky_, Apr 30 2022
%o A066829 (Python)
%o A066829 from operator import ixor
%o A066829 from functools import reduce
%o A066829 from sympy import factorint
%o A066829 def A066829(n): return reduce(ixor, factorint(n).values(),0)&1 # _Chai Wah Wu_, Jan 01 2023
%Y A066829 Characteristic function of A026424 (positions of 1's). Cf. also A028260 (its complement, positions of 0's).
%Y A066829 Cf. A001222 (bigomega), A007421, A008836, A055038 (partial sums), A065043, A069545 (run lengths), A072203, A349905, A353556, A353558, A358751, A358753.
%Y A066829 Cf. A000035.
%K A066829 nonn,easy
%O A066829 1,1
%A A066829 _G. L. Honaker, Jr._, Jan 17 2002
%E A066829 Corrected and comment added by _Reinhard Zumkeller_, Jun 26 2009