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%I #86 Dec 14 2023 08:57:01
%S 1,0,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,1,0,1,1,1,
%T 1,1,0,1,1,1,0,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,1,0,1,1,1,1,1,0,1,
%U 1,1,0,1,0,1,1,1,1,1,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,0,1,0,1,1
%N Characteristic function of nonprimes: 0 if n is prime, else 1.
%C Number of orbits of length n in map whose periodic points are A023890. - _Thomas Ward_
%C Characteristic function of nonprimes A018252. - _Jonathan Vos Post_, Dec 30 2007
%C Triangle A157423 = A005171 in every column. A052284 = INVERT transform of A005171, and the eigensequence of triangle A157423. - _Gary W. Adamson_, Feb 28 2009
%D Douglas Hofstadter, Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought.
%H Antti Karttunen, <a href="/A005171/b005171.txt">Table of n, a(n) for n = 1..65537</a>
%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H Yash Puri and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H Yash Puri and Thomas Ward, <a href="http://www.fq.math.ca/Scanned/39-5/puri.pdf">A dynamical property unique to the Lucas sequence</a>, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F a(n) = (1/n)* Sum_{ d divides n } mu(d)*A023890(n/d). E.g., a(6) = 1 since the 6th term of A023890 is 7 and the first term is 1. [edited by _Michel Marcus_, Dec 14 2023]
%F a(n) = 1 - A010051(n). - _Jonathan Vos Post_, Dec 30 2007
%F a(n) equals the first column in a table T defined by the recurrence: If n = k then T(n,k) = 1 else if k = 1 then T(n,k) = 1 - Product_{k divides n} of T(n,k), else if k divides n then T(n,k) = T(n/k,1). This is true since T(n,k) = 0 when k divides n and n/k is prime which results in Product_{k divides n} = 0 for the composite numbers and where k ranges from 2 to n. Therefore there is a remaining 1 in the expression 1-Product_{k divides n}, in the first column. Provided below is a Mathematica program as an illustration. - _Mats Granvik_, Sep 21 2013
%F a(n) = A057427(A239968(n)). - _Reinhard Zumkeller_, Mar 30 2014
%p A005171 := proc(n)
%p if isprime(n) then
%p 0 ;
%p else
%p 1 ;
%p end if;
%p end proc: # _R. J. Mathar_, May 26 2017
%t a[n_] := If[PrimeQ@ n, 0, 1]; Array[a, 105] (* _Robert G. Wilson v_, Jun 20 2011 *)
%t nn = 105; t[n_, k_] := t[n, k] = If[n == k, 1, If[k == 1, 1 - Product[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 1], 1]]; Table[t[n, 1], {n, 1, nn}] (* _Mats Granvik_, Sep 21 2013 *)
%o (PARI) a(n)=if(n<1, 0, !isprime(n)) /* _Michael Somos_, Jun 08 2005 */
%o (Haskell)
%o a005171 = (1 -) . a010051 -- _Reinhard Zumkeller_, Mar 30 2014
%o (Python)
%o from sympy import isprime
%o def a(n): return int(not isprime(n))
%o print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Oct 28 2021
%Y Cf. A010051, A018252, A023890.
%Y Cf. A157423, A157424, A052284, A050374.
%K nonn,easy
%O 1,1
%A _Russ Cox_
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