A063524 Characteristic function of 1.

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

Author

Labos Elemer, Jul 30 2001

Keywords

• easy
• nonn
• mult

Comments

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

References

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

Links

• 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

Crossrefs

Cf. A000007 (same sequence shifted once left).

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

Programs

Haskell

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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)