OFFSET
1,1
COMMENTS
Number of orbits of length n in map whose periodic points are A023890. - Thomas Ward
Characteristic function of nonprimes A018252. - Jonathan Vos Post, Dec 30 2007
REFERENCES
Douglas Hofstadter, Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
FORMULA
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]
a(n) = 1 - A010051(n). - Jonathan Vos Post, Dec 30 2007
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
MAPLE
MATHEMATICA
a[n_] := If[PrimeQ@ n, 0, 1]; Array[a, 105] (* Robert G. Wilson v, Jun 20 2011 *)
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 *)
PROG
(PARI) a(n)=if(n<1, 0, !isprime(n)) /* Michael Somos, Jun 08 2005 */
(Haskell)
a005171 = (1 -) . a010051 -- Reinhard Zumkeller, Mar 30 2014
(Python)
from sympy import isprime
def a(n): return int(not isprime(n))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Oct 28 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved