%I #121 Jan 07 2025 19:05:28

%S 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,

%T 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,

%U 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

%N Characteristic function of 1.

%C The identity function for Dirichlet multiplication (see Apostol).

%C Sum of the Moebius function mu(d) of the divisors d of n. - _Robert G. Wilson v_, Sep 30 2006

%C -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

%C a(A000012(n)) = 1; a(A087156(n)) = 0. - _Reinhard Zumkeller_, Oct 11 2008

%C 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

%C From _Christopher Hunt Gribble_, Jul 11 2013: (Start)

%C 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.

%C o--o

%C | |

%C o--o (End)

%C 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

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

%H Antti Karttunen, <a href="/A063524/b063524.txt">Table of n, a(n) for n = 0..100000</a>

%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative Functions</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F From _Philippe Deléham_, Nov 25 2008: (Start)

%F G.f.: x.

%F E.g.f.: x. (End)

%F a(n) = mu(n^2). - _Enrique Pérez Herrero_, Sep 04 2009

%F a(n) = floor(n/A000203(n)) for n > 0. - _Enrique Pérez Herrero_, Nov 11 2009

%F a(n) = (1-(-1)^(2^abs(n-1)))/2 = (1-(-1)^(2^((n-1)^2)))/2. - _Luce ETIENNE_, Jun 05 2015

%F a(n) = n*(A057427(n) - A057427(n-1)) = A000007(abs(n-1)). - _Chayim Lowen_, Aug 01 2015

%F a(n) = A010051(p*n) for any prime p (where A010051(0)=0). - _Chayim Lowen_, Aug 05 2015

%F From _Antti Karttunen_, Jun 04 2022: (Start)

%F For n >= 1:

%F 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.

%F a(n) = [A101296(n) == 1], where [ ] is the Iverson bracket.

%F Fully multiplicative with a(p^e) = 0. (End)

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

%t Table[If[n == 1, 1, 0], {n, 0, 104}] (* _Robert G. Wilson v_, Sep 30 2006 *)

%t LinearRecurrence[{1},{0,1,0},106] (* _Ray Chandler_, Jul 15 2015 *)

%o (Haskell)

%o a063524 = fromEnum . (== 1) -- _Reinhard Zumkeller_, Apr 01 2012

%o (PARI) a(n)=n==1; \\ _Charles R Greathouse IV_, Apr 01 2012

%o (Python)

%o def A063524(n): return int(n==1) # _Chai Wah Wu_, Feb 04 2022

%Y Cf. A000007 (same sequence shifted once left).

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

%K easy,nonn,mult

%O 0,1

%A _Labos Elemer_, Jul 30 2001