cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

References

  • 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.

Links

Crossrefs

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).

Programs

  • Axiom
    [moebiusMu(n) for n in 1..100]
  • Haskell
    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
  • Haskell
    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]];
  • Maple
    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
  • Mathematica
    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 *)
  • Maxima
    A008683(n):=moebius(n)$ makelist(A008683(n),n,1,30); /* Martin Ettl, Oct 24 2012 */
  • PARI
    a=n->if(n<1,0,moebius(n));
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, 1 - X)[n])};
  • PARI
    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
  • Python
    from sympy import mobius
print([mobius(i) for i in range(1, 101)])  # Indranil Ghosh, Mar 18 2017
  • Sage
    @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

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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