A055037 -id:A055037 - cp's OEIS frontend

A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Nov 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Harry J. Smith)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Characteristic function of A028260 (positions of 1's). Cf. also A026424 (positions of 0's) and A320655.
One less than A007421.
Cf. A003961, A008836, A010052, A038548 (inverse Möbius transform), A046523, A055037 (partial sums), A343784, A347102, A353337, A353338, A353555, A353557, A353629, A353669, A358750, A358752, A353374, A358775, A356163, A356170.
Cf. also A066829, A353374.

A065043 := proc(n)
    if type(numtheory[bigomega](n),'even') then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 26 2013

Table[(LiouvilleLambda[n]+1)/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)

{ for (n=1, 1000, a=1 - bigomega(n)%2; write("b065043.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 04 2009

A065043(n) = (1 - (bigomega(n)%2)); \\ Antti Karttunen, Apr 19 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A065043(n): return (reduce(ixor, factorint(n).values(),0)&1)^1 # Chai Wah Wu, Jan 01 2023

a(n) = 1 - A001222(n) mod 2.
a(n) = A007421(n) - 1.
a(n) = 1 - A066829(n).
a(A028260(k)) = 1 and a(A026424(k)) = 0 for all k.
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - Enrique Pérez Herrero, Jul 06 2012
a(n) = (A008836(n) + 1)/2. - Enrique Pérez Herrero, Jul 07 2012
a(n) = A001222(2n) mod 2. - Wesley Ivan Hurt, Jun 22 2013
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^2)/(1 - x^n). - Ilya Gutkovskiy, Apr 25 2017
From Antti Karttunen, Dec 01 2022: (Start)
For x, y >= 1, a(x*y) = 1 - abs(a(x)-a(y)).
a(n) = a(A046523(n)) = A356163(A003961(n)).
a(n) = A000035(A356163(n)+A347102(n)).
a(n) = A010052(n) + A353669(n).
a(n) = A353555(n) + A353557(n).
a(n) = A358750(n) + A358752(n).
a(n) = A353374(n) + A358775(n).
a(n) >= A356170(n).
(End)

Corrected by Charles R Greathouse IV, Sep 02 2009

A055038 Number of numbers <= n with an odd number of prime factors (counted with multiplicity).

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, 17, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 22, 23, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 30, 30, 30, 30, 31, 31, 32, 32, 33, 33, 33, 34, 35, 36, 36, 37, 38, 39, 40, 40, 41

Offset: 1

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

nonn

Partial sums of A066829.

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

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

Cf. A001222, A002819, A008836, A055037, A066829.

a055038 n = a055038_list !! (n-1)
a055038_list = scanl1 (+) a066829_list
-- Reinhard Zumkeller, Nov 19 2011

Boole[OddQ[PrimeOmega[#]]]& /@ Range[100] // Accumulate (* Jean-François Alcover, Nov 21 2019 *)

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

from operator import ixor
from functools import reduce
from sympy import factorint
def A055038(n): return sum(1 for i in range(1,n+1) if 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 Reinhard Zumkeller, Nov 19 2011

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)).

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

A212818 Numbers up to 10^n with an even number of not necessarily distinct prime factors, or positive Liouville function.

Original entry on oeis.org

1, 5, 49, 493, 4953, 49856, 499735, 4999579, 49998058, 499987392, 4999941987, 49999828888, 499999738687, 4999999516711

Offset: 0

Martin Renner, May 28 2012

nonn

			a(1) = 5 since up to 10 there are the five numbers 1, 4, 6, 9, 10 with an even number of prime factors, or positive Liouville function.

Eric Weisstein's World of Mathematics, Liouville Function

Cf. A055037 (goes up to n rather than 10^n), A002819, A008836, A028260, A065043, 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(G);

Table[Length[Select[Range[10^n], EvenQ[PrimeOmega[#]] &]], {n, 0, 5}] (* Alonso del Arte, May 28 2012 *)
Table[Count[LiouvilleLambda[Range[10^n]], 1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

a(n) = A011557(n) - A212819(n).
a(n) = (10^n)/2 + A090410(n)/2. - Donovan Johnson, May 30 2012
a(n) = A055037(10^n). - Ray Chandler, May 30 2012

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

cp's OEIS Frontend

A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

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A055038 Number of numbers <= n with an odd number of prime factors (counted with multiplicity).

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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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A212818 Numbers up to 10^n with an even number of not necessarily distinct prime factors, or positive Liouville function.

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Mathematica

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Extensions

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.