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Numbers n such that the least Fibonacci number F_k which is a multiple of n has a prime index, i.e., k is in A000040.
5

%I #54 Jul 16 2017 15:15:47

%S 2,5,13,37,73,89,113,149,157,193,233,269,277,313,353,389,397,457,557,

%T 613,673,677,733,757,877,953,977,997,1069,1093,1153,1213,1237,1453,

%U 1597,1657,1753,1873,1877,1933,1949,1993,2017,2137,2221,2237,2309,2333,2417,2473

%N Numbers n such that the least Fibonacci number F_k which is a multiple of n has a prime index, i.e., k is in A000040.

%C Numbers n such that A001177(n) is prime.

%C Each natural number n belongs to this sequence if the smallest Fibonacci number which it divides is a term of A030426. - _Jon E. Schoenfield_, Feb 28 2014

%C A092395 gives all the primes in this sequence (cf. Wikipedia-link), and the first composite occurs as the 69th term, where a(69)=4181 while A092395(69)=4273. After 4181 (= 37*113 = F_19), the next term missing from A092395 is a(148)=10877 (= 73*149. A001177(10877) = 37, F_37 = 24157817 = 2221*10877). Both of these numbers (4181 and 10877) occur in various lists of Fibonacci-related pseudoprimes. Sequence A238082 gives all composites occurring in this sequence.

%C If n is in this sequence then all divisors d > 1 of n are in this sequence. - _Charles R Greathouse IV_, Feb 04 2014

%C Composite members begin 4181, 10877, 75077, 162133, 330929, .... - _Charles R Greathouse IV_, Mar 07 2014

%H Antti Karttunen and Charles R Greathouse IV, <a href="/A233281/b233281.txt">Table of n, a(n) for n = 1..2000</a> (first 157 terms from Karttunen)

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fibonacci_prime#Divisibility_of_Fibonacci_numbers">Fibonacci prime, section: Divisibility of Fibonacci numbers</a>

%F A010051(A001177(a(n))) = 1. - _Reinhard Zumkeller_, Apr 04 2014

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define A233281 (MATCHING-POS 1 1 (lambda (n) (prime? (A001177 n)))))

%o (PARI) is(n)=my(k); while(fibonacci(k++)%n, ); isprime(k) \\ _Charles R Greathouse IV_, Feb 04 2014

%o (PARI) entry(p)=my(k=1);while(fibonacci(k++)%p,);k;

%o is(n)={

%o if(n%2==0,return(n==2));

%o if(n<13, return(n==5));

%o my(f=factor(n),p,F);

%o if(f[1,2]>1 && f[1,1]<1e14,return(0));

%o p=entry(f[1,1]);

%o F=fibonacci(p);

%o if(f[1,2]>1 && F%f[1,1]^f[1,2],return(0));

%o if(!isprime(p), return(0));

%o for(i=2,#f~,

%o if(F%f[i,1]^f[i,2],return(0))

%o );

%o 1

%o }; \\ _Charles R Greathouse IV_, Feb 04 2014

%o (Haskell)

%o a233281 n = a233281_list !! (n-1)

%o a233281_list = filter ((== 1) . a010051 . a001177) [1..]

%o -- _Reinhard Zumkeller_, Apr 04 2014

%Y Disjoint union of A092395 and A238082. The first 68 terms are identical with A092395, after which follows the first case of the latter sequence, with a(69) = A238082(1) = 4181.

%Y Cf. A000045, A001177, A001602, A030426, A051694, A060442, A086597, A233282.

%K nonn

%O 1,1

%A _Antti Karttunen_, Dec 13 2013