A007428 -id:A007428 - cp's OEIS frontend

A008683 Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1

Offset: 1

N. J. A. Sloane

Moebius inversion: f(n) = Sum_{d|n} g(d) for all n <=> g(n) = Sum_{d|n} mu(d)*f(n/d) for all n.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3, 1).
A008683 = A140579^(-1) * A140664. - Gary W. Adamson, May 20 2008
Coons & Borwein prove that Sum_{n>=1} mu(n) z^n is transcendental. - Jonathan Vos Post, Jun 11 2008; edited by Charles R Greathouse IV, Sep 06 2017
Equals row sums of triangle A144735 (the square of triangle A054533). - Gary W. Adamson, Sep 20 2008
Conjecture: a(n) is the determinant of Redheffer matrix A143104 where T(n, n) = 0. Verified for the first 50 terms. - Mats Granvik, Jul 25 2008
From Mats Granvik, Dec 06 2008: (Start)
The Editorial Office of the Journal of Number Theory kindly provided (via B. Conrey) the following proof of the conjecture: Let A be A143104 and B be A143104 where T(n, n) = 0.
"Suppose you expand det(B_n) along the bottom row. There is only a 1 in the first position and so the answer is (-1)^n times det(C_{n-1}) say, where C_{n-1} is the (n-1) by (n-1) matrix obtained from B_n by deleting the first column and the last row. Now the determinant of the Redheffer matrix is det(A_n) = M(n) where M(n) is the sum of mu(m) for 1 <= m <= n. Expanding det(A_n) along the bottom row, we see that det(A_n) = (-1)^n * det(C_{n-1}) + M(n-1). So we have det(B_n) = (-1)^n * det(C_{n-1}) = det(A_n) - M(n-1) = M(n) - M(n-1) = mu(n)." (End)
Conjecture: Consider the table A051731 and treat 1 as a divisor. Move the value in the lower right corner vertically to a divisor position in the transpose of the table and you will find that the determinant is the Moebius function. The number of permutation matrices that contribute to the Moebius function appears to be A074206. - Mats Granvik, Dec 08 2008
Convolved with A152902 = A000027, the natural numbers. - Gary W. Adamson, Dec 14 2008
[Pickover, p. 226]: "The probability that a number falls in the -1 mailbox turns out to be 3/Pi^2 - the same probability as for falling in the +1 mailbox". - Gary W. Adamson, Aug 13 2009
Let A = A176890 and B = A * A * ... * A, then the leftmost column in matrix B converges to the Moebius function. - Mats Granvik, Gary W. Adamson, Apr 28 2010 and May 28 2020
Equals row sums of triangle A176918. - Gary W. Adamson, Apr 29 2010
Calculate matrix powers: A175992^0 - A175992^1 + A175992^2 - A175992^3 + A175992^4 - ... Then the Mobius function is found in the first column. Compare this to the binomial series for (1+x)^-1 = 1 - x + x^2 - x^3 + x^4 - ... . - Mats Granvik, Gary W. Adamson, Dec 06 2010
From Richard L. Ollerton, May 08 2021: (Start)
Formulas for the numerous OEIS entries involving the Möbius transform (Dirichlet convolution of a(n) and some sequence h(n)) can be derived using the following (n >= 1):
Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
Use of gcd(n,k)*lcm(n,k) = n*k provides further variations. (End)
Formulas for products corresponding to the sums above are also available for sequences f(n) > 0: Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))). - Richard L. Ollerton, Nov 08 2021

			G.f. = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 64-65.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
Clifford A. Pickover, "The Math Book, from Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics", Sterling Publishing, 2009, p. 226. - Gary W. Adamson, Aug 13 2009
G. Pólya and G. Szegő, Problems and Theorems in Analysis Volume II. Springer_Verlag 1976.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 98-99.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 826.
All Angles, What is the Moebius function?, video, 2024.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 705-707.
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Olivier Bordellès, Some Explicit Estimates for the Mobius Function , J. Int. Seq. 18 (2015) 15.11.1
G. J. Chaitin, Thoughts on the Riemann hypothesis arXiv:math/0306042 [math.HO], 2003.
Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.
Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.
Tom Edgar, Posets and Möbius Inversion, slides, (2008).
Mats Granvik, Inverse of a triangular matrix using determinants, Inverse of a triangular matrix using matrix multiplication, Inverse of a triangular matrix as a binomial series, The ordinary generating function for the Mobius function.
Keith Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n).
A. F. Möbius, Über eine besondere Art von Umkehrung der Reihen. Journal für die reine und angewandte Mathematik 9 (1832), 105-123.
Ed Pegg Jr., The Mobius function (and squarefree numbers).
Anders Björner and Richard P. Stanley, A combinatorial miscellany.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects Of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238.
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, Jun 05 2014, pp. 105-124.
Gerard Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Wikipedia, Moebius function.
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Variants of a(n) are A178536, A181434, A181435.
Cf. A000010, A001221, A008966, A007423, A080847, A002321 (partial sums), A069158, A055615, A129360, A140579, A140664, A140254, A143104, A152902, A206706, A063524, A007427, A007428, A124010, A073776, A074206, A132971, A156552.
Cf. A059956 (Dgf at s=2), A088453 (Dgf at s=3), A215267 (Dgf at s=4), A343308 (Dgf at s=5).

Axiom
```
[moebiusMu(n) for n in 1..100]
```

import Math.NumberTheory.Primes.Factorisation (factorise)
a008683 = mu . snd . unzip . factorise where
mu [] = 1; mu (1:es) = - mu es; mu (_:es) = 0
-- Reinhard Zumkeller, Dec 13 2015, Oct 09 2013

a008683 1 = 1
a008683 n = - sum [a008683 d | d <- [1..(n-1)], n `mod` d == 0]
-- Harry Richman, Jun 13 2025

Magma
```
[ MoebiusMu(n) : n in [1..100]];
```

with(numtheory): A008683 := n->mobius(n);
with(numtheory): [ seq(mobius(n), n=1..100) ];
# Note that older versions of Maple define mobius(0) to be -1.
# This is unwise! Moebius(0) is better left undefined.
with(numtheory):
mu:= proc(n::posint) option remember; `if`(n=1, 1,
       -add(mu(d), d=divisors(n) minus {n}))
     end:
seq(mu(n), n=1..100);  # Alois P. Heinz, Aug 13 2008

Array[ MoebiusMu, 100]
(* Second program: *)
m = 100; A[_] = 0;
Do[A[x_] = x - Sum[A[x^k], {k, 2, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 20 2019, after Ilya Gutkovskiy *)

A008683(n):=moebius(n)$ makelist(A008683(n),n,1,30); /* Martin Ettl, Oct 24 2012 */

PARI
```
a=n->if(n<1,0,moebius(n));
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 - X)[n])};

list(n)=my(v=vector(n,i,1)); forprime(p=2, sqrtint(n), forstep(i=p, n, p, v[i]*=-1); forstep(i=p^2, n, p^2, v[i]=0)); forprime(p=sqrtint(n)+1, n, forstep(i=p, n, p, v[i]*=-1)); v \\ Charles R Greathouse IV, Apr 27 2012

from sympy import mobius
print([mobius(i) for i in range(1, 101)])  # Indranil Ghosh, Mar 18 2017

@cached_function
def mu(n):
    if n < 2: return n
    return -sum(mu(d) for d in divisors(n)[:-1])
# Changing the sign of the sum gives the number of ordered factorizations of n A074206.
print([mu(n) for n in (1..96)])  # Peter Luschny, Dec 26 2016

Sum_{d|n} mu(d) = 1 if n = 1 else 0.
Dirichlet generating function: Sum_{n >= 1} mu(n)/n^s = 1/zeta(s). Also Sum_{n >= 1} mu(n)*x^n/(1-x^n) = x.
In particular, Sum_{n > 0} mu(n)/n = 0. - Franklin T. Adams-Watters, Jun 20 2014
phi(n) = Sum_{d|n} mu(d)*n/d.
a(n) = A091219(A091202(n)).
Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson, Aug 01 2001
abs(a(n)) = Sum_{d|n} 2^A001221(d)*a(n/d). - Benoit Cloitre, Apr 05 2002
Sum_{d|n} (-1)^(n/d)*mobius(d) = 0 for n > 2. - Emeric Deutsch, Jan 28 2005
a(n) = (-1)^omega(n) * 0^(bigomega(n) - omega(n)) for n > 0, where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
mu(n) = A129360(n) * (1, -1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
mu(n) = -Sum_{d < n, d|n} mu(d) if n > 1 and mu(1) = 1. - Alois P. Heinz, Aug 13 2008
a(n) = A174725(n) - A174726(n). - Mats Granvik, Mar 28 2010
a(n) = first column in the matrix inverse of a triangular table with the definition: T(1, 1) = 1, n > 1: T(n, 1) is any number or sequence, k = 2: T(n, 2) = T(n, k-1) - T(n-1, k), k > 2 and n >= k: T(n,k) = (Sum_{i = 1..k-1} T(n-i, k-1)) - (Sum_{i = 1..k-1} T(n-i, k)). - Mats Granvik, Jun 12 2010
Product_{n >= 1} (1-x^n)^(-a(n)/n) = exp(x) (product form of the exponential function). - Joerg Arndt, May 13 2011
a(n) = Sum_{k=1..n, gcd(k,n)=1} exp(2*Pi*i*k/n), the sum over the primitive n-th roots of unity. See the Apostol reference, p. 48, Exercise 14 (b). - Wolfdieter Lang, Jun 13 2011
mu(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n. (conjecture). - Mats Granvik, Nov 20 2011
Sum_{k=1..n} a(k)*floor(n/k) = 1 for n >= 1. - Peter Luschny, Feb 10 2012
a(n) = floor(omega(n)/bigomega(n))*(-1)^omega(n) = floor(A001221(n)/A001222(n))*(-1)^A001221(n). - Enrique Pérez Herrero, Apr 27 2012
Multiplicative with a(p^e) = binomial(1, e) * (-1)^e. - Enrique Pérez Herrero, Jan 19 2013
G.f. A(x) satisfies: x^2/A(x) = Sum_{n>=1} A( x^(2*n)/A(x)^n ). - Paul D. Hanna, Apr 19 2016
a(n) = -A008966(n)*A008836(n)/(-1)^A005361(n) = -floor(rad(n)/n)Lambda(n)/(-1)^tau(n/rad(n)). - Anthony Browne, May 17 2016
a(n) = Kronecker delta of A001221(n) and A001222(n) (which is A008966) multiplied by A008836(n). - Eric Desbiaux, Mar 15 2017
a(n) = A132971(A156552(n)). - Antti Karttunen, May 30 2017
Conjecture: a(n) = Sum_{k>=0} (-1)^(k-1)*binomial(A001222(n)-1, k)*binomial(A001221(n)-1+k, k), for n > 1. Verified for the first 100000 terms. - Mats Granvik, Sep 08 2018
From Peter Bala, Mar 15 2019: (Start)
Sum_{n >= 1} mu(n)*x^n/(1 + x^n) = x - 2*x^2. See, for example, Pólya and Szegő, Part V111, Chap. 1, No. 71.
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 - x^n) = x + 2*(x^2 + x^4 + x^8 + x^16 + ...).
Sum_{n >= 1} (-1)^(n+1)*mu(n)*x^n/(1 + x^n) = x - 2*(x^4 + x^8 + x^16 + x^32 + ...).
Sum_{n >= 1} |mu(n)|*x^n/(1 - x^n) = Sum_{n >= 1} (2^w(n))*x^n, where w(n) is the number of different prime factors of n (Hardy and Wright, Chapter XVI, Theorem 264).
Sum_{n odd} |mu(n)|*x^n/(1 + x^(2*n)) = Sum_{n in S_1} (2^w_1(n))*x^n, where S_1 = {1, 5, 13, 17, 25, 29, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 1 (mod 4), and w_1(n) is the number of different prime factors p = 1 (mod 4) of n.
Sum_{n odd} (-1)^((n-1)/2)*mu(n)*x^n/(1 - x^(2*n)) = Sum_{n in S_3} (2^w_3(n))*x^n, where S_3 = {1, 3, 7, 9, 11, 19, 21, ...} is the multiplicative semigroup of positive integers generated by 1 and the primes p = 3 (mod 4), and where w_3(n) is the number of different prime factors p = 3 (mod 4) of n. (End)
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 11 2019
a(n) = sign(A023900(n)) * [A007947(n) = n] where [] is the Iverson bracket. - I. V. Serov, May 15 2019
a(n) = Sum_{k = 1..n} gcd(k, n)*a(gcd(k, n)) = Sum_{d divides n} a(d)*d*phi(n/d). - Peter Bala, Jan 16 2024

A063524 Characteristic function of 1.

Original entry on oeis.org

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

Offset: 0

Labos Elemer, Jul 30 2001

The identity function for Dirichlet multiplication (see Apostol).
Sum of the Moebius function mu(d) of the divisors d of n. - Robert G. Wilson v, Sep 30 2006
-a(n) is the Hankel transform of A000045(n), n >= 0 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - Wolfdieter Lang, Jan 23 2007
a(A000012(n)) = 1; a(A087156(n)) = 0. - Reinhard Zumkeller, Oct 11 2008
a(n) for n >= 1 is the Dirichlet convolution of following functions b(n), c(n), a(n) = Sum_{d|n} b(d)*c(n/d): a(n) = A008683(n) * A000012(n), a(n) = A007427(n) * A000005(n), a(n) = A007428(n) * A007425(n). - Jaroslav Krizek, Mar 03 2009
From Christopher Hunt Gribble, Jul 11 2013: (Start)
a(n) for 1 <= n <= 4 and conjectured for n > 4 is the number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element: When n=1, there is only 1 Hamiltonian circuit in a 2 X 2 square lattice, as illustrated below.  The circuit is the same when rotated and/or reflected and so has only 1 orbital element under the symmetry group of the square.
     o--o
     |  |
     o--o  (End)
Convolution property: For any sequence b(n), the sequence c(n)=b(n)*a(n) has the following values: c(1)=0, c(n+1)=b(n) for all n > 1. In other words, the sequence b(n) is shifted 1 step to the right. - David Neil McGrath, Nov 10 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

Antti Karttunen, Table of n, a(n) for n = 0..100000
G. P. Michon, Multiplicative Functions
Wikipedia, Dirichlet convolution
Index entries for sequences computed from exponents in factorization of n
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for characteristic functions

Cf. A000007 (same sequence shifted once left).

Cf. A000005, A000010, A000012, A000027, A003763, A008683, A007427, A007428, A007425, A023900, A055615, A101296, A227005, A227257, A227301, A209077.

a063524 = fromEnum . (== 1)  -- Reinhard Zumkeller, Apr 01 2012

A063524 := proc(n) if n = 1 then 1 else 0; fi; end;

Table[If[n == 1, 1, 0], {n, 0, 104}] (* Robert G. Wilson v, Sep 30 2006 *)
LinearRecurrence[{1},{0,1,0},106] (* Ray Chandler, Jul 15 2015 *)

a(n)=n==1; \\ Charles R Greathouse IV, Apr 01 2012

def A063524(n): return int(n==1) # Chai Wah Wu, Feb 04 2022

From Philippe Deléham, Nov 25 2008: (Start)
G.f.: x.
E.g.f.: x. (End)
a(n) = mu(n^2). - Enrique Pérez Herrero, Sep 04 2009
a(n) = floor(n/A000203(n)) for n > 0. - Enrique Pérez Herrero, Nov 11 2009
a(n) = (1-(-1)^(2^abs(n-1)))/2 = (1-(-1)^(2^((n-1)^2)))/2. - Luce ETIENNE, Jun 05 2015
a(n) = n*(A057427(n) - A057427(n-1)) = A000007(abs(n-1)). - Chayim Lowen, Aug 01 2015
a(n) = A010051(p*n) for any prime p (where A010051(0)=0). - Chayim Lowen, Aug 05 2015
From Antti Karttunen, Jun 04 2022: (Start)
For n >= 1:
a(n) = Sum_{d|n} A000010(n/d) * A023900(d), and similarly for any pair of sequences that are Dirichlet inverses of each other, like for example A000027 & A055615 and those mentioned in Krizek's Mar 03 2009 comment above.
a(n) = [A101296(n) == 1], where [ ] is the Iverson bracket.
Fully multiplicative with a(p^e) = 0. (End)

A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 3, 10, 6, 9, 3, 18, 3, 9, 9, 15, 3, 18, 3, 18, 9, 9, 3, 30, 6, 9, 10, 18, 3, 27, 3, 21, 9, 9, 9, 36, 3, 9, 9, 30, 3, 27, 3, 18, 18, 9, 3, 45, 6, 18, 9, 18, 3, 30, 9, 30, 9, 9, 3, 54, 3, 9, 18, 28, 9, 27, 3, 18, 9, 27, 3, 60, 3, 9, 18, 18, 9, 27, 3, 45, 15, 9, 3, 54, 9, 9, 9, 30, 3

Offset: 1

N. J. A. Sloane, May 24 1994

Let n = Product p_i^e_i. Tau (A000005) is tau_2, this sequence is tau_3, A007426 is tau_4, where tau_k(n) (also written as d_k(n)) = Product_i binomial(k-1+e_i, k-1) is the k-th Piltz function. It gives the number of ordered factorizations of n as a product of k terms. - Len Smiley
Inverse Möbius transform applied twice to all 1's sequence.
A085782 gives the range of values of this sequence. - Matthew Vandermast, Jul 12 2004
Appears to equal the number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. - Wouter Meeussen, Sep 11 2004
Number of divisors of n's divisors. - Lekraj Beedassy, Sep 07 2004
Number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. If the partition is not a box, there is a minimal i+j where b_{i,j} != b_{1,1} and an element can be added there. - Franklin T. Adams-Watters, Jun 14 2006
Equals row sums of A127170. - Gary W. Adamson, May 20 2007
Equals A134577 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 02 2007
Equals row sums of triangle A143354. - Gary W. Adamson, Aug 10 2008
a(n) is congruent to 1 (mod 3) if n is a perfect cube, otherwise a(n) is congruent to 0 (mod 3). - Geoffrey Critzer, Mar 20 2015
Also row sums of A195050. - Omar E. Pol, Nov 26 2015
Number of 3D grids of n congruent boxes with three different edge lengths, in a box, modulo rotation (cf. A034836 for cubes instead of boxes and A140773 for boxes with two different edge lengths; cf. A000005 for the 2D case). - Manfred Boergens, Apr 06 2021
Number of ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			a(6) = 9; the divisors of 6 are {1,2,3,6} and the numbers of divisors of these divisors are 1, 2, 2, and 4. Adding them, we get 9 as a result.
Also, since 6 is a squarefree number, the formula from Herrero can be used to obtain the result: a(6) = 3^omega(6) = 3^2 = 9. - _Wesley Ivan Hurt_, May 30 2014

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
R. Eisinga, R. Breitling, and T. Heskes, The exact probability distribution of the rank product statistics for replicated experiments, FEBS Letters, 2013, 587: 677-682, doi:10.1016/j.febslet.2013.01.037
Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, and Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
N. J. A. Sloane, Transforms.
Qingfeng Sun and Deyu Zhang, Sums of the triple divisor function over values of a ternary quadratic form, arXiv:1510.06170 [math.NT], 2015.
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.
Wikipedia, Adolf Piltz.

Cf. A000005 (Mobius transform), A007426 (inverse Mobius transform), A061201 (partial sums), A127270, A143354, A027750, A007428 (Dirichlet inverse), A175596.
Column k=3 of A077592.
Additional cross-references mentioned in a comment: A034836, A038548, A140733.

a007425 = sum . map a000005 . a027750_row
-- Reinhard Zumkeller, Feb 16 2012

f:=proc(n) local t1,i,j,k; t1:=0; for i from 1 to n do for j from 1 to n do for k from 1 to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;
A007425 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(2+e[j][2],2), j=1..nops(e)); end; # Len Smiley

f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Sep 13 2004 *)
SetAttributes[tau, Listable]; tau[1, n_] := 1; tau[k_, n_] := Plus @@ (tau[k-1, Divisors[n]]); Table[tau[3, n], {n, 100}] (* Enrique Pérez Herrero, Nov 08 2009 *)
Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 50}] (* Wesley Ivan Hurt, May 30 2014 *)
f[p_, e_] := (e+1)*(e+2)/2;  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 27 2019 *)

for(n=1,100,print1(sumdiv(n,k,numdiv(k)),","))

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3)[n]) \\ Ralf Stephan

a(n)=sumdiv(n, x, sumdiv(x, y, 1 )) \\ Joerg Arndt, Oct 07 2012

a(n)=sumdivmult(n,k,numdiv(k)) \\ Charles R Greathouse IV, Aug 30 2013

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^3)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

from math import prod, comb
from sympy import factorint
def A007425(n): return prod(comb(2+e,2) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

a(n) = Sum_{d dividing n} tau(d). - Benoit Cloitre, Apr 04 2002
G.f.: Sum_{k>=1} tau(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
For n = Product p_i^e_i, a(n) = Product_i A000217(e_i + 1). - Lekraj Beedassy, Sep 07 2004
Dirichlet g.f.: zeta^3(s).
From Enrique Pérez Herrero, Nov 03 2009: (Start)
a(n^2) = tau_3(n^2) = tau_2(n^2)*tau_2(n), where tau_2 is A000005 and tau_3 is this sequence.
a(s) = 3^omega(s), if s>1 is squarefree (A005117) and omega(s) is: A001221. (End)
From Enrique Pérez Herrero, Nov 08 2009: (Start)
a(n) = tau_3(n) = tau_2(n)*tau_2(n*rad(n))/tau_2(rad(n)), where rad(n) is A007947 and tau_2(n) is A000005.
tau_3(n) >= 2*tau_2(n) - 1.
tau_3(n) <= tau_2(n)^2 + tau_2(n)-1. (End)
From Vladimir Shevelev, Dec 22 2017: (Start)
a(n) = sqrt(Sum_{d|n}(tau(d))^3);
a(n) = |Sum_{d|n} A008836(d)*(tau(d))^2|.
The first formula follows from the first Cloitre formula and a Liouville formula; the second formula follows from our analogous formula (cf. our comment in Formula section of A000005). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

A007427 Moebius transform applied twice to sequence 1,0,0,0,....

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -2, 0, 1, 4, -2, -2, -2, 4, 4, 0, -2, -2, -2, -2, 4, 4, -2, 0, 1, 4, 0, -2, -2, -8, -2, 0, 4, 4, 4, 1, -2, 4, 4, 0, -2, -8, -2, -2, -2, 4, -2, 0, 1, -2, 4, -2, -2, 0, 4, 0, 4, 4, -2, 4, -2, 4, -2, 0, 4, -8, -2, -2, 4, -8, -2, 0, -2, 4, -2, -2, 4, -8, -2, 0, 0

Offset: 1

N. J. A. Sloane

|a(n)| is the number of ways to write n as a product of 2 squarefree numbers (i.e., number of ways to write n = x*y with 1 <= x <= n, 1 <= y <= n, x and y squarefree). - Benoit Cloitre, Jan 01 2003

			G.f. = x - 2*x^2 - 2*x^3 + x^4 - 2*x^5 + 4*x^6 - 2*x^7 + x^9 + 4*x^10 + ...
We have a(3^1) = C(2, 1)*(-1)^1 = -2, a(3^2) = C(2, 2)*(-1)^2 = 1, and a(3^m) = C(2, m)*(-1)^m = 0 for m >= 3. - _Petros Hadjicostas_, Jun 07 2019

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series, arXiv:2205.03680 [math.CO], 2022.
Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.
Enrique Pérez Herrero, Mathematica Package for Piltz Divisor Functions.
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
N. J. A. Sloane, Transforms.
Wikipedia, Adolf Piltz.

Dirichlet inverse of A000005, Mobius transform of A008683.
Cf. A063524, A007428, A005117, A008836, A046099, A225817.
Cf. A013661, A001620, A306016.

a007427 n = sum $ zipWith (*) mds $ reverse mds where
   mds = a225817_row n
-- Reinhard Zumkeller, Jul 30 2013

möbius := proc(a)  local b, i, mo: b := NULL:
mo := (m,n) -> `if`(irem(m,n) = 0, numtheory:-mobius(m/n), 0);
for i to nops(a) do b := b, add(mo(i,j)*a[j], j=1..i) od: [b] end:
(möbius@@2)([1, seq(0, i=1..80)]); # Peter Luschny, Sep 08 2017

f[n_] := Plus @@ Times @@@ (MoebiusMu[{#, n/#}] & /@ Divisors@n); Array[f, 105] (* Robert G. Wilson v *)
a[n_] := DivisorSum[n, MoebiusMu[#]*MoebiusMu[n/#]&]; Array[a, 80] (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( n<1, 0, direuler(p=2, n, (1 - X)^2)[n])}; /* Michael Somos, Nov 15 2002 */

{a(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d)))}; /* Michael Somos, Nov 15 2002 */

a(n)=if(n>1,my(f=factor(n)[,2],s=sum(i=1,#f,f[i]==1));if(vecmax(f)>2,0,(-1)^s<Charles R Greathouse IV, Jun 28 2011

from math import prod, comb
from sympy import factorint
def A007427(n): return prod(-comb(2,e) if e&1 else comb(2,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

Dirichlet g.f.: 1/zeta(s)^2.
Multiplicative function with a(p^e) = binomial(2, e)*(-1)^e for p prime and e >= 0.
a(n) = Sum_{d|n} mu(d)*mu(n/d). - Benoit Cloitre, Apr 05 2002
a(n^2) = A008683(n)^2. a(A005117(n)) = (-2)^A001221(A005117(n)). - Enrique Pérez Herrero, Jun 27 2011 [Misrendering of contribution rectified by Peter Munn, Mar 06 2020]
a(n) is the Dirichlet inverse of A000005, which means a(n) = -Sum_{d|n, dA000005(n/d)*a(d). - Enrique Pérez Herrero, Jan 19 2013
a(n) = 0 if n is not cubefree: A046099, otherwise sign(a(n)) = lambda(n), where lambda is A008836. - Enrique Pérez Herrero, Jan 19 2013
Dirichlet g.f. of |a(n)|: zeta(s)^2/zeta(2s)^2 (conjectured). - Ralf Stephan, Jul 05 2013.  The conjecture is correct because 1+Sum_{e>=1} binomial(2,e)/p^(e*s) = (p^s+1)^2/p^2s, whose product over p is zeta(s)^2/zeta(2s)^2. - Michael Shamos
a(n) = Sum_{k=1..A000005(n)} A225817(n,k)*A225817(n,n+1-k). - Reinhard Zumkeller, Jul 30 2013
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} tau(k)*A(x^k), where tau = A000005. - Ilya Gutkovskiy, May 11 2019
Sum_{k=1..n} abs(a(k)) ~ (n/zeta(2)^2) * (log(n) + 2*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 24 2023

Added a proof of Stephan's conjecture about the Dirichlet g.f. of |a(n)|.

A007431 a(n) = Sum_{d|n} phi(d)*mu(n/d).

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 0, 5, 2, 4, 0, 9, 1, 11, 0, 3, 4, 15, 0, 17, 3, 5, 0, 21, 2, 16, 0, 12, 5, 27, 0, 29, 8, 9, 0, 15, 4, 35, 0, 11, 6, 39, 0, 41, 9, 12, 0, 45, 4, 36, 0, 15, 11, 51, 0, 27, 10, 17, 0, 57, 3, 59, 0, 20, 16, 33, 0, 65, 15, 21, 0, 69, 8, 71, 0, 16, 17, 45, 0, 77, 12, 36, 0, 81, 5, 45, 0

Offset: 0

N. J. A. Sloane

Also Moebius transform applied twice to natural numbers.
Also number of complex primitive Dirichlet characters modulo n and Sum_{k=1..n} a(k) is asymptotic to (18/Pi^4)*n^2. - Steven Finch, Feb 16 2006
Dirichlet convolution of phi(n) and mu(n). - Richard L. Ollerton, May 07 2021
From Jianing Song, May 21 2022: (Start)
a(n) is the number of degree-psi(n) primitive Dirichlet characters mod n, where psi = A002322. Also, a(n) is the number of degree-(k*psi(n)) primitive Dirichlet characters mod n for all k >= 1.
a(n) is the maximum element in the n-th row of A354058 (or A354061). (End)

			From _Jianing Song_, May 21 2022: (Start)
a(45) = 12: psi(45) = 12, there are 3 degree-12 primitive characters modulo 5 and 4 degree-12 primitive characters modulo 9, so a(45) = 3 * 4 = 12.
a(63) = 20: psi(63) = 6, there are 5 sextic primitive characters modulo 7 and 4 sextic primitive characters modulo 9, so a(63) = 5 * 4 = 20. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 0..20000 (terms 0..1000 from T. D. Noe)
H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
N. J. A. Sloane, Transforms

Cf. A007432.

Cf. A000010, A008683, A130054 (Dirichlet inverse), A354058, A354061.

a007431 0 = 0
a007431 n = sum $ map (a008683 . gcd n) [1..n]
-- Reinhard Zumkeller, Jan 06 2014

[0] cat [&+[EulerPhi(d)*MoebiusMu(Floor(n/d)):d in Divisors(n)]:n in [1..90]]; // Marius A. Burtea, Aug 10 2019

with(numtheory); f:=n->add( phi(d)*mobius(n/d), d in divisors(n)); [seq(f(n),n=0..120)];

Table[Sum[EulerPhi[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 0, 86}] (* Jean-François Alcover, Apr 04 2011 *)
Table[DirichletConvolve[MoebiusMu[n], EulerPhi[n], n, m], {m, 86}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := If[e == 1, p-2, p^e - 2*p^(e-1) + p^(e-2)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 23 2020 *)

a(n)=if(n<1,0,direuler(p=2,n,(1-X)^2/(1-p*X))[n]) \\ Ralf Stephan

a(n) = sumdiv(n,d, moebius(d) * eulerphi(n/d) ); \\ Joerg Arndt, Apr 14 2013

A007431(n) = if(!n,n,my(f=factor(n)); prod(i=1, #f~, if(1==f[i, 2], f[i, 1]-2, ((f[i,1]-1)^2)*(f[i, 1]^(f[i, 2]-2))))); \\ Antti Karttunen, Dec 15 2024, after Vladeta Jovovic's multiplicative formula

Multiplicative with a(p) = p-2 and a(p^e) = (p-1)^2*p^(e-2) for e > 1. - Vladeta Jovovic, Jan 25 2002
Dirichlet g.f.: zeta(s-1)/zeta^2(s).
a(n) = Sum_{k=1..n} mu(gcd(n,k)) for n > 0. - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{k=1..n} (phi(gcd(k,n)) * cos(2*Pi*k/n)). - Enrique Pérez Herrero, Jan 18 2013
a(n) = Sum_{d|n} tau_{-2}(d)*n/d = Sum_{d|n} tau_{-3}(d)*sigma_1(n/d), where tau_{-3} is A007428, tau_{-2} A007427 and sigma_1 A000203. - Enrique Pérez Herrero, Jan 19 2013
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2. - Ilya Gutkovskiy, Apr 25 2017
Sum_{k=1..n} a(k) ~ 18 * n^2 / Pi^4. - Vaclav Kotesovec, Nov 04 2018
Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} phi(n)*x^n. - Mamuka Jibladze, Aug 09 2019
Sum_{d|n} a(d) = phi(n) (A000010). - Amiram Eldar, Jun 23 2020
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = A354058(n,psi(n)) = A354061(n,psi(n)) with psi = A002322. - Jianing Song, May 21 2022

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A008683 Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.

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A063524 Characteristic function of 1.

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A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.

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A007427 Moebius transform applied twice to sequence 1,0,0,0,....

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A007431 a(n) = Sum_{d|n} phi(d)*mu(n/d).

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