A174863 -id:A174863 - cp's OEIS frontend

A224987 Numbers such that Liouville's function (A002819) and the little omega analog to Liouville's function (A174863) are equal.

1, 2, 3, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 13092, 13093, 13094, 13095, 13096, 13097, 13098, 13099, 13100, 13101, 13102, 13103, 13104, 13105, 13106, 13107, 13232, 13233, 13234, 13235, 13239, 13240, 13241, 13242

Offset: 1

Donovan Johnson, Apr 22 2013

nonn

Numbers n such that A002819(n) = A174863(n). There are 9056 terms <= 10^12 (the largest is 16959554). For n from 16959555 to 10^12, A002819(n) < A174863(n).

			n = 43:
A002819(n) = sum_{k = 1..n} (-1)^bigomega(k) = -3.
A174863(n) = sum_{k = 1..n} (-1)^omega(k) = -3.
A002819(43) = A174863(43) = -3.

Donovan Johnson, Table of n, a(n) for n = 1..9056

Cf. A001221, A001222, A008836, A002819, A174863.

PrimeOmega[n_] := Plus @@ FactorInteger[n][[All, 2]]; PrimeNu[n_] := Length[FactorInteger[n]]; Reap[For[s1 = 0; s2 = 0; n = 1, n < 15000, n++, s1 = s1 + (-1)^PrimeOmega[n]; s2 = s2 + (-1)^PrimeNu[n]; If[s1 == s2, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, May 03 2013, after Pari *)

s1=0; s2=0; c=0; for(n=1, 16959554, s1=s1+(-1)^bigomega(n); s2=s2+(-1)^omega(n); if(s1==s2, c++; write("b224987.txt", c " " n)))

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

Original entry on oeis.org

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, -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, -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

Offset: 1

Reinhard Zumkeller, Oct 14 2002

Multiplicative: a(1) = 1, a(n) for n >=2 is sign of parity of number of distinct primes dividing n. a(p) = -1, a(pq) = 1, a(pq...z) = (-1)^k, a(p^k) = -1, where p,q,.. z are distinct primes and k natural numbers. - Jaroslav Krizek, Mar 17 2009

a(n) is the unitary Moebius function, i.e., the inverse of the constant 1 function under the unitary convolution defined by (f X g)(n)= sum of f(d)g(n/d), where the sum is over the unitary divisors d of n (divisors d of n such that gcd(d,n/d)=1). - Laszlo Toth, Oct 08 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.

Cf. A174863 (partial sums).

a076479 = a008683 . a007947  -- Reinhard Zumkeller, Jun 01 2013

[(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018

A076479 := proc(n)
    (-1)^A001221(n) ;
end proc:
seq(A076479(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

Table[(-1)^PrimeNu[n], {n,50}] (* Enrique Pérez Herrero, Jan 17 2013 *)

N=66;
mu=vector(N); mu[1]=1;
{ for (n=2,N,
    s = 0;
    fordiv (n,d,
        if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
        s += mu[d];
    );
    mu[n] = -s;
); };
mu /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */

a(n)=(-1)^omega(n) \\ Charles R Greathouse IV, Aug 02 2013

from math import prod
from sympy.ntheory import mobius, primefactors
def A076479(n): return mobius(prod(primefactors(n))) # Chai Wah Wu, Sep 24 2021

a(n) = A008683(A007947(n)).
a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002
a(n) = sign(A180403(n)). - Mats Granvik, Oct 08 2010
Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - R. J. Mathar, Apr 11 2011
Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - R. J. Mathar, Apr 19 2011
Sum_{n>=1} a(n)/n^2 = A065469. - R. J. Mathar, Apr 19 2011
a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - Enrique Pérez Herrero, Jan 17 2013
a(A030230(n)) = -1; a(A030231(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - Álvar Ibeas, Dec 30 2018
Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))*rad(n/gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - Ridouane Oudra, May 25 2023

A346617 Irregular triangle T(n,m) read by rows (n >= 1, 1 <= m <= Max(A001221([1..n]))): T(n,m) = number of integers in [1,n] with m distinct prime factors.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 1, 5, 1, 6, 1, 7, 1, 7, 2, 8, 2, 8, 3, 9, 3, 9, 4, 9, 5, 10, 5, 11, 5, 11, 6, 12, 6, 12, 7, 12, 8, 12, 9, 13, 9, 13, 10, 14, 10, 14, 11, 15, 11, 15, 12, 16, 12, 16, 12, 1, 17, 12, 1, 18, 12, 1, 18, 13, 1, 18, 14, 1, 18, 15, 1, 18, 16, 1, 19, 16, 1, 19, 17, 1

Offset: 1

N. J. A. Sloane, Aug 19 2021

Column k >= 1 of the triangle gives the number of numbers i in the range 1 <= i <= n with omega(i) = A001221(i) = k.

A285577 is a similar triangle which has an extra column on the left for k = 0.

			Rows 1 through 12 are:
1 [0]
2 [1]
3 [2]
4 [3]
5 [4]
6 [4, 1]
7 [5, 1]
8 [6, 1]
9 [7, 1]
10 [7, 2]
11 [8, 2]
12 [8, 3]
13 [9, 3]

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 52-56.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, Vol. 1, p. 211, Eq. (5).

Alois P. Heinz, Rows n = 1..10000, flattened
N. J. A. Sloane, The first 100 rows.

Cf. A001221, A013939, A069811, A123066, A174863, A285577.

Row lengths give A111972 (for n>1).

omega := proc(n) nops(numtheory[factorset](n)) end proc: # # A001221
A:=Array(1..20,0);
ans:=[[0]];
mx:=0;
for n from 2 to 100 do
k:=omega(n);
if k>mx then mx:=k; fi;
A[k]:=A[k]+1;
ans:=[op(ans),[seq(A[i],i=1..mx)]];
od:
ans;
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 0,
      b(n-1)+x^nops(ifactors(n)[2]))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..max(1, degree(p))))(b(n)):
seq(T(n), n=1..40);  # Alois P. Heinz, Aug 19 2021

T[n_] := If[n == 1, {0},
     Range[n] // PrimeNu // Tally // Rest // #[[All, 2]]&];
Array[T, 40] // Flatten (* Jean-François Alcover, Mar 08 2022 *)

For fixed k, T(n,k) ~ (1/(k-1)!) * n * (log log n)^(k-1) / log n [Landau].
From Alois P. Heinz, Aug 19 2021: (Start)
Sum_{k>=1} k * T(n,k) = A013939(n).
Sum_{k>=1} k^2 * T(n,k) = A069811(n).
Sum_{k>=1} (-1)^(k-1) * T(n,k) = A123066(n).
Sum_{k>=1} (-1)^k * T(n,k) = -1 + A174863(n).
Sum_{k>=1} T(n,k) = n - 1. (End)

A275547 Numbers n that have an equal number of even and odd values of A001221(k) for 1 <= k <= n.

Original entry on oeis.org

2, 40, 46, 48, 50, 7960, 7962, 7984, 7986, 8808, 8810, 8812, 8816, 8822, 8824, 8826, 8828, 8830, 8836, 8844, 8846, 8848, 8850, 8854, 8856, 8858, 8860, 8862, 8864, 8866, 8872, 8878, 8970, 8972, 8974, 9064, 9078, 9080, 9082, 9084, 9086, 9088, 9096, 9164, 9220

Offset: 1

G. L. Honaker, Jr., Aug 01 2016

nonn

Is this sequence infinite?

			a(2) = 40 because if we check omega(n) = A001221(n) for each n = 1..40, then half will be even numbers and half will be odd numbers.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A001221, A005408, A005843, A174863.

omega:= n-> nops(numtheory[factorset](n)):
b:= proc(n) option remember; (-1)^omega(n)+`if`(n>1, b(n-1), 0) end:
a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while b(k)<>0 do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 02 2016

a[1] = 2; a[n_] := a[n] = Block[{k = a[n-1], s=0}, While[(s += (-1)^ PrimeNu[++k]) != 0]; k]; Array[a, 100] (* Giovanni Resta, Aug 03 2016 *)

is(n) = my(i=0, j=0); for(k=1, n, if(omega(k)%2==0, i++, j++)); if(i==j, return(1), return(0)) \\ Felix Fröhlich, Aug 02 2016

isok(n) = {my(v = vector(n, k, omega(k))); #select(x->x % 2 == 1, v) == n/2;} \\ Michel Marcus, Aug 02 2016

A174863(a(n)) = 0. - Alois P. Heinz, Aug 02 2016

More terms from Alois P. Heinz, Aug 02 2016

A346457 a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.

Original entry on oeis.org

1, 4, 5, 8, 9, 32, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Robert Israel, Table of n, a(n) for n = 1..3223

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346456.

N:= 10000: # for values <= N
omega:= n -> nops(numtheory:-factorset(n)):
R:= map(n -> (-1)^omega(n),[$1..10000]):
S:= map(abs,ListTools:-PartialSums(R)):
m:= max(S):
V:= Vector(m):
for i from 1 to N do if S[i] > 0 and V[S[i]] = 0 then V[S[i]]:= i fi od:
convert(V,list); # Robert Israel, Oct 30 2023

Table[k=1;While[Abs[Sum[(-1)^PrimeNu@j,{j,k}]]!=n,k++];k,{n,30}] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (abs(sum(j=1, k, (-1)^omega(j))) != n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : |Sum_{j=1..k} mu(rad(j))| = n}, where mu is the Moebius function and rad is the squarefree kernel.

A327666 a(n) = Sum_{k = 1..n} (-1)^(Omega(k) - omega(k)), where Omega(k) counts prime factors of k with multiplicity and omega(k) counts distinct prime factors.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

nonn

Partial sums of A162511.

			Omega(1) = omega(1) = 0. The difference is 0, so (-1)^0 = 1, so a(1) = 1.
Omega(2) = omega(2) = 1. The difference is 0, so (-1)^0 = 1, which is added to a(1) to give a(2) = 2.
Omega(3) = omega(3) = 1. The difference is 0, so (-1)^0 = 1, which is added to a(2) to give a(3) = 3.
Omega(4) = 2 but omega(4) = 1. The difference is 1, so (-1)^1 = -1, which is added to a(3) to give a(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A002321, A002819, A008836, A046660, A069812, A076479, A162511, A174863, A307868.

Table[Sum[(-1)^(PrimeOmega[k] - PrimeNu[k]), {k, n}], {n, 70}]
f[p_, e_] := (-1)^(e - 1); Accumulate @ Table[Times @@ f @@@ FactorInteger[n], {n, 1, 100}] (* Amiram Eldar, Sep 18 2022 *)

seq(n)={my(v=vector(n)); v[1]=1; for(k=2, n, v[k] = v[k-1] + (-1)^(bigomega(k)-omega(k))); v} \\ Andrew Howroyd, Sep 23 2019

from functools import reduce
from sympy import factorint
def A327666(n): return sum(-1 if reduce(lambda a,b:~(a^b), factorint(i).values(),0)&1 else 1 for i in range(1,n+1)) # Chai Wah Wu, Jan 01 2023

a(1) = 1, a(n) = a(n - 1) + (-1)^(Omega(n) - omega(n)) for n > 1.

a(n) ~ c * n, where c = A307868. - Amiram Eldar, Sep 18 2022

A346455 a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = n, where omega(j) is the number of distinct primes dividing j.

Original entry on oeis.org

1, 52, 55, 56, 57, 58, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346456, A346457.

a[n_]:=(k=1;While[Sum[(-1)^PrimeNu@j,{j,k}]!=n,k++];k);Array[a,25] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) !=n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = n}, where mu is the Moebius function and rad is the squarefree kernel.

A346456 a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = -n, where omega(j) is the number of distinct primes dividing j.

Original entry on oeis.org

3, 4, 5, 8, 9, 32, 9283, 9284, 9285, 9292, 9293, 9294, 9295, 9296, 9343, 9434, 9437, 9440, 9479, 9686, 9689, 9690, 9697, 9698, 9699, 9700, 9711, 9716, 9717, 9718, 9719, 9720, 9721, 9740, 9741, 9852, 9855, 9856, 9857, 10284, 10285, 10286, 10305, 10314, 10325, 10326, 10331, 10338

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346457.

a[n_]:=(k=1;While[Sum[(-1)^PrimeNu@j,{j,k}]!=-n,k++];k);Array[a,6] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) != -n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = -n}, where mu is the Moebius function and rad is the squarefree kernel.

A368405 Infinitary version of Mertens's function: a(n) = Sum_{k=1..n} A064179(k).

Original entry on oeis.org

1, 0, -1, -2, -3, -2, -3, -2, -3, -2, -3, -2, -3, -2, -1, -2, -3, -2, -3, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 1, 2, 3, 2, 1, 0, -1, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 5

Offset: 1

Amiram Eldar, Dec 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rasa Steuding, Jörn Steuding, and László Tóth, A modified Möbius mu-function, Rendiconti del Circolo Matematico di Palermo, Vol. 60 (2011), pp. 13-21; arXiv preprint, arXiv:1109.4242 [math.NT], 2011.

Partial sums of A064179.

Similar sequences: A002321, A174863 (unitary), A209802 (exponential).

f[p_, e_] := (-1)^DigitCount[e, 2, 1]; imu[1] = 1; imu[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[imu, 100]]

imu(n) = vecprod(apply(x -> (-1)^hammingweight(x), factor(n)[, 2]));
lista(nmax) = {my(s = 0); for(k = 1, nmax, s+ = imu(k); print1(s, ", "));}

A300270 a(n) = Sum_{1 <= i <= j <= n} mu(ij)floor((n/i)/j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 20, 20, 20, 21, 22, 23, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 32, 32, 32, 33, 34, 34, 35, 35, 35, 36, 37, 37, 37, 38, 38, 39, 40, 41, 42, 42, 42, 43, 43, 44, 44, 44, 45, 46, 47, 48, 48, 48

Offset: 1

Benoit Cloitre, Mar 01 2018

nonn

We have Sum_{k=1..n} mu(k)*floor(n/k) = 1 and lim_{n -> infinity} Sum_{1 <= i <= j <= n} (mu(i*j)/i)/j = 1/2.

Cf. A174863, A008683.

a(n) = sum(j=1, n, sum(i=1, j, moebius(i*j)*floor(n/i/j)))

a(n) ~ n/2 (n->infinity).

a(n) = (A174863(n) + n)/2. - Ridouane Oudra, May 16 2025

cp's OEIS Frontend

A224987 Numbers such that Liouville's function (A002819) and the little omega analog to Liouville's function (A174863) are equal.

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A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

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A346617 Irregular triangle T(n,m) read by rows (n >= 1, 1 <= m <= Max(A001221([1..n]))): T(n,m) = number of integers in [1,n] with m distinct prime factors.

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A275547 Numbers n that have an equal number of even and odd values of A001221(k) for 1 <= k <= n.

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A346457 a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.

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A327666 a(n) = Sum_{k = 1..n} (-1)^(Omega(k) - omega(k)), where Omega(k) counts prime factors of k with multiplicity and omega(k) counts distinct prime factors.

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A300270 a(n) = Sum_{1 <= i <= j <= n} mu(ij)floor((n/i)/j).