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A063524 Characteristic function of 1.

%I A063524 #121 Jan 07 2025 19:05:28
%S A063524 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 A063524 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 A063524 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 A063524 Characteristic function of 1.
%C A063524 The identity function for Dirichlet multiplication (see Apostol).
%C A063524 Sum of the Moebius function mu(d) of the divisors d of n. - _Robert G. Wilson v_, Sep 30 2006
%C A063524 -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 A063524 a(A000012(n)) = 1; a(A087156(n)) = 0. - _Reinhard Zumkeller_, Oct 11 2008
%C A063524 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 A063524 From _Christopher Hunt Gribble_, Jul 11 2013: (Start)
%C A063524 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 A063524      o--o
%C A063524      |  |
%C A063524      o--o  (End)
%C A063524 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 A063524 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
%H A063524 Antti Karttunen, <a href="/A063524/b063524.txt">Table of n, a(n) for n = 0..100000</a>
%H A063524 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative Functions</a>
%H A063524 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>
%H A063524 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%H A063524 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%H A063524 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A063524 From _Philippe Deléham_, Nov 25 2008: (Start)
%F A063524 G.f.: x.
%F A063524 E.g.f.: x. (End)
%F A063524 a(n) = mu(n^2). - _Enrique Pérez Herrero_, Sep 04 2009
%F A063524 a(n) = floor(n/A000203(n)) for n > 0. - _Enrique Pérez Herrero_, Nov 11 2009
%F A063524 a(n) = (1-(-1)^(2^abs(n-1)))/2 = (1-(-1)^(2^((n-1)^2)))/2. - _Luce ETIENNE_, Jun 05 2015
%F A063524 a(n) = n*(A057427(n) - A057427(n-1)) = A000007(abs(n-1)). - _Chayim Lowen_, Aug 01 2015
%F A063524 a(n) = A010051(p*n) for any prime p (where A010051(0)=0). - _Chayim Lowen_, Aug 05 2015
%F A063524 From _Antti Karttunen_, Jun 04 2022: (Start)
%F A063524 For n >= 1:
%F A063524 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 A063524 a(n) = [A101296(n) == 1], where [ ] is the Iverson bracket.
%F A063524 Fully multiplicative with a(p^e) = 0. (End)
%p A063524 A063524 := proc(n) if n = 1 then 1 else 0; fi; end;
%t A063524 Table[If[n == 1, 1, 0], {n, 0, 104}] (* _Robert G. Wilson v_, Sep 30 2006 *)
%t A063524 LinearRecurrence[{1},{0,1,0},106] (* _Ray Chandler_, Jul 15 2015 *)
%o A063524 (Haskell)
%o A063524 a063524 = fromEnum . (== 1)  -- _Reinhard Zumkeller_, Apr 01 2012
%o A063524 (PARI) a(n)=n==1; \\ _Charles R Greathouse IV_, Apr 01 2012
%o A063524 (Python)
%o A063524 def A063524(n): return int(n==1) # _Chai Wah Wu_, Feb 04 2022
%Y A063524 Cf. A000007 (same sequence shifted once left).
%Y A063524 Cf. A000005, A000010, A000012, A000027, A003763, A008683, A007427, A007428, A007425, A023900, A055615, A101296, A227005, A227257, A227301, A209077.
%K A063524 easy,nonn,mult
%O A063524 0,1
%A A063524 _Labos Elemer_, Jul 30 2001