A347328 -id:A347328 - cp's OEIS frontend

A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

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, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1

Offset: 1

N. J. A. Sloane

Coons and Borwein: "We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non-trivial completely multiplicative function from N to {-1,1}, the series sum_{n=1..infinity} f(n)z^n is transcendental over {Z}[z]; in particular, sum_{n=1..infinity} lambda(n)z^n is transcendental, where lambda is Liouville's function. The transcendence of sum_{n=1..infinity} mu(n)z^n is also proved." - Jonathan Vos Post, Jun 11 2008
Coons proves that a(n) is not k-automatic for any k > 2. - Jonathan Vos Post, Oct 22 2008
The Riemann hypothesis is equivalent to the statement that for every fixed epsilon > 0, lim_{n -> infinity} (a(1) + a(2) + ... + a(n))/n^(1/2 + epsilon) = 0 (Borwein et al., theorem 1.2). - Arkadiusz Wesolowski, Oct 08 2013

			a(4) = 1 because since bigomega(4) = 2 (the prime divisor 2 is counted twice), then (-1)^2 = 1.
a(5) = -1 because 5 is prime and therefore bigomega(5) = 1 and (-1)^1 = -1.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 1-11.
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.
P. Ribenboim, Algebraic Numbers, p. 44.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.3.5 on page 99.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 112.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Borwein, R. Ferguson, and M. J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.
Benoit Cloitre, A tauberian approach to RH, arXiv:1107.0812 [math.NT], 2011.
Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.
Michael Coons, (Non)Automaticity of number theoretic functions, arXiv:0810.3709 [math.NT], 2008.
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]
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Andrei Vieru, Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions, arXiv:1306.0496 [math.GM], 2015.
H. Walum, A recurrent pattern in the list of quadratic residues mod a prime and in the values of the Liouville lambda function, J. Numb. Theory 12 (1) (1980) 53-56.
Eric Weisstein's World of Mathematics, Liouville Function
Wikipedia, Liouville function
Index to divisibility sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A002053, A007421, A002819 (partial sums), A008683, A010052, A026424, A028260, A028488, A056912, A056913, A065043, A066829, A106400, A156552, A349905, A063647, A101455, A048691.
Möbius transform of A010052.
Cf. A182448 (Dgf at s=2), A347328 (Dgf at s=3), A347329 (Dgf at s=4).

a008836 = (1 -) . (* 2) . a066829  -- Reinhard Zumkeller, Nov 19 2011

A008836 := n -> (-1)^numtheory[bigomega](n); # Peter Luschny, Sep 15 2011
with(numtheory): A008836 := proc(n) local i,it,s; it := ifactors(n): s := (-1)^add(it[2][i][2], i=1..nops(it[2])): RETURN(s) end:

Table[LiouvilleLambda[n], {n, 100}] (* Enrique Pérez Herrero, Dec 28 2009 *)
Table[If[OddQ[PrimeOmega[n]],-1,1],{n,110}] (* Harvey P. Dale, Sep 10 2014 *)

{a(n) = if( n<1, 0, n=factor(n); (-1)^sum(i=1, matsize(n)[1], n[i,2]))}; /* Michael Somos, Jan 01 2006 */

a(n)=(-1)^bigomega(n) \\ Charles R Greathouse IV, Jan 09 2013

from sympy import factorint
def A008836(n): return -1 if sum(factorint(n).values()) % 2 else 1 # Chai Wah Wu, May 24 2022

Dirichlet g.f.: zeta(2s)/zeta(s); Dirichlet inverse of A008966.
Sum_{ d divides n } lambda(d) = 1 if n is a square, otherwise 0.
Completely multiplicative with a(p) = -1, p prime.
a(n) = (-1)^A001222(n) = (-1)^bigomega(n). - Jonathan Vos Post, Apr 16 2006
a(n) = A033999(A001222(n)). - Jaroslav Krizek, Sep 28 2009
Sum_{d|n} a(d) *(A000005(d))^2 = a(n) *Sum{d|n} A000005(d). - Vladimir Shevelev, May 22 2010
a(n) = 1 - 2*A066829(n). - Reinhard Zumkeller, Nov 19 2011
a(n) = i^(tau(n^2)-1) where tau(n) is A000005 and i is the imaginary unit. - Anthony Browne, May 11 2016
a(n) = A106400(A156552(n)). - Antti Karttunen, May 30 2017
Recurrence: a(1)=1, n > 1: a(n) = sign(1/2 - Sum_{dMats Granvik, Oct 11 2017
a(n) = Sum_{ d | n } A008683(d)*A010052(n/d). - Jinyuan Wang, Apr 20 2019
a(1) = 1; a(n) = -Sum_{d|n, d < n} mu(n/d)^2 * a(d). - Ilya Gutkovskiy, Mar 10 2021
a(n) = (-1)^A349905(n). - Antti Karttunen, Apr 26 2022
From Ridouane Oudra, Jun 02 2024: (Start)
a(n) = (-1)^A066829(n);
a(n) = (-1)^A063647(n);
a(n) = A101455(A048691(n));
a(n) = sin(tau(n^2)*Pi/2). (End)

A182448 Decimal expansion of Pi^2/15.

Original entry on oeis.org

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

Offset: 0

Mats Granvik, Apr 29 2012

			0.65797362673929...

Thomas L. Curtright, Bernoulli Partitions, arXiv:2502.09633 [math.CO], 2025. See p. 3. Mentions this sequence.
László Tóth, Linear combinations of Dirichlet series associated with the Thue-Morse sequence, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A98; arXiv preprint, arXiv:2211.13570 [math.NT], 2022.
Index entries for transcendental numbers.

Cf. A000796, A008836, A010060, A013661, A013662, A086463, A191898, A249103.

Cf. A347328, A347329, A347330, A347331.

RealDigits[N[Sum[1/(n + 0)^2 - 1/(n + 1)^2 + 1/(n + 2)^2 - 1/(n + 3)^2 - 4/(n + 4)^2 - 1/(n + 5)^2 + 1/(n + 6)^2 - 1/(n + 7)^2 + 1/(n + 8)^2 + 4/(n + 9)^2, {n, 1, Infinity, 10}], 90]][[1]]
RealDigits[N[Sum[LiouvilleLambda[n]/n^2, {n, 1, Infinity}], 90]][[1]]
RealDigits[Pi^2/15,10,120][[1]] (* Harvey P. Dale, May 28 2017 *)

PARI
```
Pi^2/15 \\ Michel Marcus, Oct 21 2014
```

See Mathematica code.
Equals Gamma(4)*zeta(4)/Pi^2 = zeta(4)/zeta(2) = A013662/A013661 = Product_{p prime} (p^2/(p^2+1)). - Stanislav Sykora, Oct 21 2014
Equals (1/10) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/3)^2 - 1/(n + 2/3)^2 ). - Peter Bala, Oct 31 2019
Equals Sum_{k>=1} A008836(k)/k^2. - Amiram Eldar, Jun 23 2020
Equals (1/10) * Sum_{k>=1} (5*t(k-1) + 3*t(k))/k^2, where t(k) = A010060(k) (Tóth, 2022). - Amiram Eldar, Feb 04 2024
Equals 3/5 + (1/5) * Sum_{n>=1} 1/(n^2*(n+1)^2). - Davide Rotondo, May 28 2025
Equals 1/A082020 = A164102/30 = A195055/5. - Hugo Pfoertner, May 28 2025

Offset corrected and more terms added by Rick L. Shepherd, Jan 08 2014

A347329 Decimal expansion of Pi^4/105.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.9277056288952613070137174541781439166640...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A008836, A182448, A347328, A347330, A347331.

Cf. A013662, A013666.

RealDigits[Pi^4/105, 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals zeta(8) / zeta(4).
Equals Sum_{k>=1} A008836(k) / k^4.
Equals Product_{p prime} 1/(1+p^(-4)). [corrected by Amiram Eldar, Jun 06 2023]

A347330 Decimal expansion of zeta(10) / zeta(5).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.96534649609163603621377296424322122474050...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A008836, A182448, A347328, A347329, A347331.

Cf. A013663, A013668.

RealDigits[Zeta[10] / Zeta[5], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals Sum_{k>=1} A008836(k) / k^5.
Equals Product_{p prime} 1/(1+p^(-5)). [corrected by Amiram Eldar, Jun 06 2023]
Equals 1/A157291. - R. J. Mathar, Jul 20 2025

A347331 Decimal expansion of 691 * Pi^6 / 675675.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 26 2021

			0.9831944836800760217380865587210155031...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A008836, A182448, A347328, A347329, A347330.

Cf. A013664, A013670.

RealDigits[691 * Pi^6 / 675675, 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals zeta(12) / zeta(6).
Equals Sum_{k>=1} A008836(k) / k^6.
Equals Product_{p prime} 1/(1+p^(-6)). [corrected by Amiram Eldar, Jun 06 2023]

Data corrected by Amiram Eldar, Jun 06 2023

A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

Original entry on oeis.org

1, 25, 217, 793, 3001, 5425, 16465, 25369, 52705, 75025, 159721, 172081, 369097, 411625, 651217, 811801, 1414945, 1317625, 2469241, 2379793, 3572905, 3993025, 6424177, 5505073, 9378001, 9227425, 12807289, 13056745, 20486761, 16280425, 28599361, 25977625, 34659457

Offset: 1

Seiichi Manyama, May 18 2024

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

Cf. A084218, A350156.
Cf. A068970, A084220, A372964.
Cf. A001160, A008683.
Cf. A002117, A013664, A347328.

f[p_, e_] := (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 5));

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_5(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1).
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(3) = 0.846335... (A347328). (End)
Dirichlet convolution of A334659 and A001160. - R. J. Mathar, Jul 14 2025

A159253 a(n) is the smallest positive integer not yet in the sequence such that n * a(n) is a cube.

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 8, 3, 100, 121, 18, 169, 196, 225, 32, 289, 12, 361, 50, 441, 484, 529, 72, 5, 676, 27, 98, 841, 900, 961, 16, 1089, 1156, 1225, 6, 1369, 1444, 1521, 200, 1681, 1764, 1849, 242, 75, 2116, 2209, 288, 7, 20, 2601, 338, 2809, 108, 3025, 392

Offset: 1

Franklin T. Adams-Watters, Apr 07 2009

This is a self-inverse permutation of the positive integers.

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

Cf. A064429 (function on exponents)

Cf. A330596, A347328.

f[p_, e_] := If[(r = Mod[e, 3]) == 0, p^e, p^(e - (-1)^r)]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n) = {my(f = factor(n), r); prod(i = 1, #f~, r=f[i,2]%3; f[i,1]^if(r == 0, f[i,2], f[i,2]-(-1)^r));} \\ Amiram Eldar, Dec 01 2022

Multiplicative with a(p^(3*n)) = p^(3*n), a(p^(3*n+1)) = p^(3*n+2), and a(p^(3*n+2)) = p^(3*n+1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A347328 * A330596 / 3 = 0.2111705... . - Amiram Eldar, Dec 01 2022

A376742 Decimal expansion of Product_{p prime} (p^3 + 1)/(p^3 - 1).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Oct 03 2024

			1.420308303489193353248184427065490...

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986. See equation 1.2.8 at p. 5.

Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 408-409.

Cf. A000578, A001221, A002117, A013664, A034444, A092732, A183030.

RealDigits[Zeta[3]^2/Zeta[6],10,100][[1]]

PARI
```
prodeulerrat((p^3 + 1)/(p^3 - 1))
```

Equals zeta(3)^2/zeta(6) = Sum_{k>=1} 2^omega(k)/k^3. See Titchmarsh and Shamos.
Equals 945*zeta(3)^2/Pi^6.
Equals A157289 / A088453 = A013664 / A347328^2. - R. J. Mathar, Jul 14 2025

cp's OEIS Frontend

A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

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A182448 Decimal expansion of Pi^2/15.

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A347329 Decimal expansion of Pi^4/105.

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A347330 Decimal expansion of zeta(10) / zeta(5).

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A347331 Decimal expansion of 691 * Pi^6 / 675675.

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A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

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A159253 a(n) is the smallest positive integer not yet in the sequence such that n * a(n) is a cube.

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