A055038 -id:A055038 - 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.

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

A055037 Number of numbers <= n with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 25, 26, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 34

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 01 2000

nonn

Partial sums of A065043.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Polya Conjecture

Cf. A001222, A002819, A008836, A055038, A065043.

Table[Length[Select[Range[n], EvenQ[PrimeOmega[#]] &]], {n, 75}] (* Alonso del Arte, May 28 2012 *)
Accumulate[Table[(LiouvilleLambda[n] + 1)/2, {n, 1, 100}]] (* Vaclav Kotesovec, Aug 18 2025 *)

first(n)=my(s); vector(n,k,s+=1-bigomega(k)%2) \\ Charles R Greathouse IV, Sep 02 2015

from functools import reduce
from operator import ixor
from sympy import factorint
def A055037(n): return sum(1 for i in range(1,n+1) if not (reduce(ixor, factorint(i).values(),0)&1)) # Chai Wah Wu, Jan 01 2023

a(n) = (1/2)*Sum_{k=1..n} (1+lambda(k)) = (1/2)*(n+L(n)), where lambda(n)=A008836(n) and L(n)=A002819(n).

Formula and more terms from Vladeta Jovovic, Dec 03 2001

Offset corrected by Ray Chandler, May 30 2012

A244152 Self-inverse permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = A028260(1+a(k)), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = A026424(a(k)).

Original entry on oeis.org

1, 4, 10, 2, 24, 7, 6, 55, 18, 3, 16, 15, 121, 44, 12, 11, 39, 9, 36, 35, 105, 31, 250, 5, 29, 28, 93, 26, 25, 86, 22, 82, 238, 79, 20, 19, 81, 72, 17, 68, 218, 65, 517, 14, 62, 67, 60, 202, 195, 57, 59, 56, 185, 477, 8, 52, 50, 175, 51, 47, 177, 45, 495, 167, 42, 161, 46, 40, 162, 169, 150, 38, 143, 455, 459, 140, 153, 1060, 34, 134, 37, 32

Offset: 1

Antti Karttunen, Jul 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A008836, A026424, A028260, A055037, A055038, A066829.

Similar entanglement permutations: A245603-A245604, A235491, A236854, A243347, A244319.

a(1) = 1, and for n > 1, if A066829(n) = 1, then a(n) = A028260(1 + A244152(A055038(n))), otherwise a(n) = A026424(A244152(A055037(n)-1)).

For all n > 1, A008836(a(n)) = -1 * A008836(n), where A008836 is Liouville's lambda-function.

A245603 Permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2a(k), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = 1+(2a(k)).

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 6, 16, 9, 7, 10, 12, 32, 17, 11, 13, 18, 14, 20, 24, 33, 19, 64, 15, 21, 25, 34, 22, 26, 36, 28, 40, 65, 35, 23, 27, 48, 37, 29, 41, 66, 38, 128, 30, 42, 49, 50, 68, 67, 44, 39, 52, 72, 129, 31, 43, 51, 69, 56, 45, 80, 53, 130, 73, 57, 70, 46, 54, 81, 96, 74, 58, 82, 131, 132, 76, 71, 256, 60

Offset: 1

Antti Karttunen, Jul 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245604.
Cf. A026424, A028260, A066829, A055037, A055038.
Similar permutations: A143692, A244152, A244321, A245613, A245605, A245607.

a(1) = 1, and for n > 1, if A066829(n) = 1, then a(n) = 2 * A245603(A055038(n)), otherwise a(n) = 1 + (2 * A245603(A055037(n)-1)).
As a composition of related permutations:
a(n) = A244321(A245613(n)).
For all n >= 1, A000035(a(n)) = 1 - A066829(n). [Permutation A143692 has the same property.]

A245613 Permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = A244991(a(k)), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = A244990(1+a(k)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 9, 7, 11, 10, 32, 18, 13, 12, 17, 15, 22, 20, 35, 19, 66, 14, 24, 21, 34, 25, 23, 33, 31, 45, 63, 37, 27, 26, 41, 36, 29, 43, 69, 40, 134, 30, 47, 39, 44, 68, 71, 50, 38, 46, 67, 131, 28, 49, 42, 70, 64, 52, 92, 48, 127, 65, 61, 75, 55, 51, 89, 83, 73, 60

Offset: 1

Antti Karttunen, Jul 27 2014

nonn

This shares with the permutation A122111 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245614.
Cf. A026424, A028260, A055037, A055038, A066829, A244990, A244991, A244992, A122111.
Similar entanglement permutations: A243287, A243343, A243345, A244321-A244322, A245603, A135141, A237427, A245605.

a(1) = 1, and for n > 1, if A066829(n) = 1, a(n) = A244991(a(A055038(n))), otherwise a(n) = A244990(1+a(A055037(n)-1)).
As a composition of related permutations:
a(n) = A244322(A245603(n)).
For all n >= 1, A066829(n) = A244992(a(n)).

A373114 Cardinality of the largest subset of {1,...,n} such that no odd number of terms from this subset multiply to a square.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 20, 20, 21, 21, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, 31, 31, 32, 33, 34, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 46, 47

Offset: 1

Terence Tao, May 25 2024

			For n=6, {2,3,5} is the largest set without an odd product being a square, so a(6)=3.

Martin Ehrenstein, Table of n, a(n) for n = 1..550 (terms 72..181 calculated from A360659)
A. Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. Math. 153 (2001), 407-470; preprint, arXiv:math/9909190 [math.NT], 1999.
Terence Tao, On product representations of squares, arXiv:2405.11610 [math.NT], May 2024.

Closely related to A360659, A372306, A373119, A373178, A373195.

Cf. A055038, A246849.

F(n, b)={vector(n, k, my(f=factor(k)); prod(i=1, #f~, if(bittest(b, primepi(f[i, 1])-1), 1, -1)^f[i, 2]))}
a(n)={my(m=oo); for(b=0, 2^primepi(n)-1, m=min(m, vecsum(F(n, b)))); (n-m)/2} \\ adapted from Andrew Howroyd, Feb 16 2023 at A360659 by David A. Corneth, May 25 2024

import itertools
import sympy
def generate_all_completely_multiplicative_functions(primes):
    combinations = list(itertools.product([-1, 1], repeat=len(primes)))
    functions = []
    for combination in combinations:
        func = dict(zip(primes, combination))
        functions.append(func)
    return functions
def evaluate_function(f, n):
    if n == 1:
        return 1
    factors = sympy.factorint(n)
    value = 1
    for prime, exp in factors.items():
        value *= f[prime] ** exp
    return value
def compute_minimum_sum(N: int):
    primes = list(sympy.primerange(1, N + 1))
    functions = generate_all_completely_multiplicative_functions(primes)
    min_sum = float("inf")
    for func in functions:
        total_sum = 0
        for n in range(1, N + 1):
            total_sum += evaluate_function(func, n)
        if total_sum < min_sum:
            min_sum = total_sum
    return min_sum
results = [(N - compute_minimum_sum(N)) // 2 for N in range(1, 12)]
print(", ".join(map(str, results)))

from itertools import product
from sympy import primerange, primepi, factorint
def A373114(n):
    a = dict(zip(primerange(n+1),range(c:=primepi(n))))
    return n-min(sum(sum(e for p,e in factorint(m).items() if b[a[p]])&1^1 for m in range(1,n+1)) for b in product((0,1),repeat=c)) # Chai Wah Wu, May 31 2024

n-2*a(n) = A360659(n) (see Footnote 2 of the linked paper of Tao).
Asymptotically, a(n)/n converges to log(1+sqrt(e)) - 2*Integral_{t=1..sqrt(e)} log(t)/(t+1) dt = A246849 ~ 0.828499... (essentially due to Granville and Soundararajan).
a(n+1)-a(n) is either 0 or 1 for any n.
a(n) >= A055038(n).

More terms from David A. Corneth, May 25 2024 using b-file from A360659 and formula n-2*a(n) = A360659(n)

A143692 Permutation of natural numbers: If n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2k, otherwise, when n is k-th number with an even number of prime divisors [i.e., n = A028260(k)], a(n) = (2k)-1.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 8, 10, 7, 9, 12, 14, 16, 11, 13, 15, 18, 20, 22, 24, 17, 19, 26, 21, 23, 25, 28, 30, 32, 34, 36, 38, 27, 29, 31, 33, 40, 35, 37, 39, 42, 44, 46, 48, 50, 41, 52, 54, 43, 56, 45, 58, 60, 47, 49, 51, 53, 55, 62, 57, 64, 59, 66, 61, 63, 68, 70, 72, 65, 74, 76, 78

Offset: 1

Reinhard Zumkeller, Aug 29 2008

nonn

a(a(n)) = A143694(n).

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A143691.

Cf. A000035, A055037, A055038, A066829, A143694, A245603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a243692 = (+ 1) . fromJust . (`elemIndex` a143691_list)
-- Reinhard Zumkeller, Aug 07 2014

N:= 1000: # to get a(1) to a(N)
Odds,Evens:= selectremove(t -> numtheory:-bigomega(t)::odd,[$1..N]):
for k from 1 to nops(Odds) do A[Odds[k]]:= 2*k od:
for k from 1 to nops(Evens) do A[Evens[k]]:= 2*k-1 od:
seq(A[k],k=1..N); # Robert Israel, Jul 27 2014

m = 100;
odds = Select[Range[m], OddQ[PrimeOmega[#]]&];
evens = Select[Range[m], EvenQ[PrimeOmega[#]]&];
Do[a[odds[[k]]] = 2k, {k, 1, Length[odds]}];
Do[a[evens[[k]]] = 2k-1, {k, 1, Length[evens]}];
Array[a, m] (* Jean-François Alcover, Mar 09 2019, from Maple *)

From Antti Karttunen, Jul 27 2014: (Start)
If A066829(n) = 1, then a(n) = 2 * A055038(n), otherwise a(n) = (2 * A055037(n)) - 1.
For all n >= 1, A000035(a(n)) = 1 - A066829(n). [Permutation A245603 has the same property].
(End)

Name changed by Antti Karttunen, Jul 27 2014

A212819 Numbers up to 10^n with an odd number of prime factors, or negative Liouville function.

Original entry on oeis.org

0, 5, 51, 507, 5047, 50144, 500265, 5000421, 50001942, 500012608, 5000058013, 50000171112, 500000261313, 5000000483289

Offset: 0

Martin Renner, May 28 2012

nonn

			a(1) = 5 since up to 10 there are the five numbers 2, 3, 5, 7, 8 with an odd number of prime factors or negative Liouville function.

Eric Weisstein's World of Mathematics, Liouville Function

Cf. A002819, A008836, A026424, A066829, A090410.

zg:=0: zu:=0: G:=[]: U:=[]: k:=0:
for i from 1 to 10^8 do if numtheory[bigomega](i) mod 2 = 0 then zg:=zg+1: else zu:=zu+1: fi: if i=10^k then G:=[op(G),zg]: U:=[op(U),zu]: k:=k+1: fi: od:
print(U);

Table[Count[LiouvilleLambda[Range[10^n]], -1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

a(n) = A011557(n) - A212818(n).
a(n) = (10^n)/2 - A090410(n)/2. - Donovan Johnson, May 30 2012
a(n) = A055038(10^n). - Ray Chandler, May 30 2012

a(9)-a(13) from Donovan Johnson, May 30 2012

A384210 Number of numbers <= n of the form p * m^2, where p is a prime and m is an integer >= 1.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 13, 13, 13, 14, 15, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 32, 33, 33, 33, 34, 35, 36, 36, 37

Offset: 1

Ridouane Oudra, May 22 2025

nonn

Partial sums of A358769.
First differs from A055038 at a(30).
a(A229125(n)) = n.

Ridouane Oudra, Table of n, a(n) for n = 1..10000

Cf. A229125, A358769, A000720, A008836, A001221.

with(numtheory): A358769:=n-> add(nops(factorset(d)), d in divisors(n)) mod 2:
seq(add(A358769(i), i=1..n), n=1..100);

a(n) = sum(k=1, n, isprime(core(k))); \\ Michel Marcus, May 29 2025

from math import isqrt
from sympy import primepi
def A384210(n): return sum(primepi(n//y**2) for y in range(1,isqrt(n)+1)) # Chai Wah Wu, Jun 06 2025

a(n) = Sum_{i=1..n} A358769(i).
a(n) = Sum_{i=1..floor(sqrt(n))} pi(floor(n/i^2)), where pi = A000720.
a(n) = - Sum_{i=1..n} lambda(i)*omega(i)*floor(n/i), where lambda = A008836 and omega = A001221.

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.

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A055037 Number of numbers <= n with an even number of prime factors (counted with multiplicity).

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A373114 Cardinality of the largest subset of {1,...,n} such that no odd number of terms from this subset multiply to a square.

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A143692 Permutation of natural numbers: If n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2*k, otherwise, when n is k-th number with an even number of prime divisors [i.e., n = A028260(k)], a(n) = (2*k)-1.

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A212819 Numbers up to 10^n with an odd number of prime factors, or negative Liouville function.

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A143692 Permutation of natural numbers: If n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2k, otherwise, when n is k-th number with an even number of prime divisors [i.e., n = A028260(k)], a(n) = (2k)-1.