A072203 -id:A072203 - cp's OEIS frontend

A380049 Partial sums of A072203.

0, 1, 3, 4, 6, 7, 9, 12, 14, 15, 17, 20, 24, 27, 29, 30, 32, 35, 39, 44, 48, 51, 55, 58, 60, 61, 63, 66, 70, 75, 81, 88, 94, 99, 103, 106, 110, 113, 115, 116, 118, 121, 125, 130, 136, 141, 147, 154, 160, 167, 173, 180, 188, 195, 201, 206, 210, 213, 217, 220, 224, 227, 231, 234, 236

Offset: 1

Tsuyoshi Hanatate, Jan 10 2025

nonn

Vaclav Kotesovec, Plot of a(n) / (2*n^(3/2)/(-3*zeta(1/2))) for n = 1..1000000

Cf. A058933, A072203.

Accumulate[Accumulate[Table[-LiouvilleLambda[n], {n, 2, 100}]]] (* Vaclav Kotesovec, Jan 15 2025 *)

f(n) = 1 - sum(i=1, n, (-1)^bigomega(i)); \\ A072203
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, Feb 06 2025

a(n) = Sum_{k=1..n} A072203(k).

Conjecture: The average value of a(n) is 2*n^(3/2)/(-3*zeta(1/2)). - Vaclav Kotesovec, Jan 15 2025

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.

Original entry on oeis.org

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, 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, 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

Offset: 1

G. L. Honaker, Jr., Jan 17 2002

From Reinhard Zumkeller, Jul 01 2009: (Start)
The first N Terms are constructed by the following sieving process:
for j:=1 until N do a(j):=0,
for i:=1 until N/2 do
for j:=2*i step i until N do a(j):=1-a(i). (End)
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

			From _Reinhard Zumkeller_, Jul 01 2009: (Start)
Sieve for N = 30, also demonstrating the affinity to the Sieve of Eratosthenes:
[initial] a(j):=0, 1<=j<=30:
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
[i=1] a(1)=0 --> a(j):=1, 2<=j<=30:
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
[i=2] a(2)=1 --> a(2*j):=0, 2<=j<=[30/2]:
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
[i=3] a(3)=1 --> a(3*j):=0, 2<=j<=[30/3]:
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
[i=4] a(4)=0 --> a(4*j):=1, 2<=j<=[30/4]:
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
[i=5] a(5)=1 --> a(5*j):=0, 2<=j<=[30/5]:
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
[i=6] a(6)=0 --> a(6*j):=1, 2<=j<=[30/6]:
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
[i=7] a(7)=1 --> a(7*j):=0, 2<=j<=[30/7]:
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
[i=8] a(8)=1 --> a(8*j):=0, 2<=j<=[30/8]:
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
[i=9] a(9)=0 --> a(9*j):=1, 2<=j<=[30/9]:
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
[i=10] a(10)=0 --> a(10*j):=1, 2<=j<=[30/10]:
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
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)

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21 (provides Dirichlet g.f.)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences generated by sieves

Characteristic function of A026424 (positions of 1's). Cf. also A028260 (its complement, positions of 0's).
Cf. A001222 (bigomega), A007421, A008836, A055038 (partial sums), A065043, A069545 (run lengths), A072203, A349905, A353556, A353558, A358751, A358753.
Cf. A000035.

a066829 = (`mod` 2) . a001222 -- Reinhard Zumkeller, Nov 19 2011

A066829 := proc(n)
    modp(numtheory[bigomega](n) ,2) ;
end proc:
seq(A066829(n),n=1..80) ; # R. J. Mathar, Jul 15 2017

Table[(1-LiouvilleLambda[n])/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)
Table[If[OddQ[PrimeOmega[n]],1,0],{n,120}] (* Harvey P. Dale, Mar 12 2016 *)

A066829(n) = (bigomega(n)%2); \\ Simplified by Antti Karttunen, Apr 30 2022

from sympy import primeomega as Omega
def a(n): return Omega(n)%2
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Apr 30 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A066829(n): return reduce(ixor, factorint(n).values(),0)&1 # Chai Wah Wu, Jan 01 2023

a(A026424(n)) = 1; a(A028260(n)) = 0.
Dirichlet g.f.: (zeta(s)^2 - zeta(2*s)) / (2*zeta(s)). [Typo corrected by Vaclav Kotesovec, Jan 30 2024]
a(n) = (1-A008836(n)) / 2. - Corrected by Antti Karttunen, Apr 30 2022
a(m*n) = a(m) XOR a(n). - Reinhard Zumkeller, Aug 28 2008
a(n) = A001222(n) mod 2. - Reinhard Zumkeller, Nov 19 2011
From Antti Karttunen, May 01 & Nov 30 2022: (Start)
a(n) = 1 - A065043(n) = A349905(n) mod 2.
a(n) = A353556(n) + A353558(n).
a(n) = A358751(n) + A358753(n). (End)
a(n) = A000035(A001222(n)). - Omar E. Pol, Apr 09 2025

Corrected and comment added by Reinhard Zumkeller, Jun 26 2009

A002053 a(n) = least value of m for which Liouville's function A002819(m) = -n.

Original entry on oeis.org

2, 3, 8, 13, 20, 31, 32, 53, 76, 79, 80, 117, 176, 181, 182, 193, 200, 283, 284, 285, 286, 293, 440, 443, 468, 661, 678, 683, 684, 1075, 1076, 1087, 1088, 1091, 1092, 1093, 1106, 1109, 1128, 1129, 1130, 1131, 1132, 1637, 1638, 1753, 1756, 1759, 1760, 2699

Offset: 0

N. J. A. Sloane

Also when n first appears in A072203(m).

H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..2000 from T. D. Noe)
H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]
H. Gupta, A table of values of Liouville's function L(n), Research Bulletin of East Panjab University, No. 3 (Feb. 1950), 45-55. [Annotated scanned copy]

Cf. A008836, A028488, A072203.

f[n_] := f[n] = f[n - 1] -(-1)^Length[Flatten[Table[ #[[1]], {#[[2]]}] & /@ FactorInteger[n]]]; f[1] = 0; Do[k = 1; While[f[k] != n, k++ ]; Print[k], {n, 1, 50}]

from functools import reduce
from operator import ixor
from itertools import count
from sympy import factorint
def A002053(n): return next(filter(lambda m:-n==sum(-1 if reduce(ixor, factorint(i).values(),0)&1 else 1 for i in range(1,m+1)),count(1))) # Chai Wah Wu, Jan 01 2023

More terms from Jud McCranie

A123066 (Number of numbers <= n with an odd number of distinct prime factors) - (number of numbers <= n with an even number of distinct prime factors).

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Sep 26 2006

Analog of A072203 for number of distinct factors. Conjecture that sequence changes sign infinitely often, although the next sign change is probably large.

The signs first change at n = 52 and then change again at n = 7954. - Harvey P. Dale, Jul 04 2012

G. C. Greubel, Table of n, a(n) for n = 1..5000
H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

Cf. A030230, A030231.
Cf. A072203, A001221.
Cf. A346617.

a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+
      `if`(nops(ifactors(n)[2])::odd, 1, -1))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Dec 21 2018

dpf[n_] := Module[{df = PrimeNu[n]}, If[OddQ[df], 1, -1]]; Join[{0}, Accumulate[ Array[dpf, 100, 2]]] (* Harvey P. Dale, Jul 04 2012 *)

from sympy import primenu
def A123066(n): return 1+sum(1 if primenu(i)&1 else -1 for i in range(1,n+1)) # Chai Wah Wu, Dec 31 2022

a(n) = Sum_{k>=1} (-1)^(k-1) * A346617(n,k). - Alois P. Heinz, Aug 19 2021

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A380049 Partial sums of A072203.

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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.

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A002053 a(n) = least value of m for which Liouville's function A002819(m) = -n.

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A123066 (Number of numbers <= n with an odd number of distinct prime factors) - (number of numbers <= n with an even number of distinct prime factors).

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