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A220292 Nonprime n not divisible by 2 or 3 such that Fibonacci(n-1) is congruent to (1 - Legendre(n,5))/2 modulo n. 1
1, 143, 1763, 1891, 4181, 5183, 5777, 6601, 6721, 8149, 10403, 10877, 13201, 13981, 15251, 17119, 17711, 30889, 34561, 36863, 40501, 51841, 64079, 64681, 67861, 68101, 68251, 75077, 78409, 79523, 88601, 88831, 90061, 96049, 97343, 97921 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A Fibonacci based primality criterion of Legendre and Lagrange which is listed as theorem 2.2 in the Zhi-Hong Sun link at A000032 states the Fibonacci(p-1) mod p = (1 - Legendre(p/5))/2. This sequence lists the pseudoprimes to this criterion which are not divisible by 2 or 3.

The number of pseudoprimes appears to decrease as n increases, there being 36 between 1 and 100,000, 17 between 100,000 and 200,000,and 11 between 200,000 and 300,000.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Zhi-Hong Sun, Congruences for Fibonacci Numbers [PDF] (Lecture notes, 2009)

MAPLE

with(numtheory): with(combinat): for n from 1 to 40000 do if n mod 2 <> 0 and n mod 3 <>0 and fibonacci(n-1) mod n = (1-legendre(n, 5))/2 and not isprime(n) then print(n) fi od;

PROG

(PARI) is(n)=gcd(n, 6)==1 && ((Mod([1, 1; 1, 0], n))^(n-1))[1, 2]==(1-kronecker(n, 5))/2 && !isprime(n) \\ Charles R Greathouse IV, Dec 22 2012

CROSSREFS

Sequence in context: A035304 A241924 A185514 * A159054 A135946 A217534

Adjacent sequences:  A220289 A220290 A220291 * A220293 A220294 A220295

KEYWORD

nonn

AUTHOR

Gary Detlefs, Dec 09 2012

EXTENSIONS

Missing a(30) added by Charles R Greathouse IV, Dec 25 2012

Name corrected by Jianing Song, Sep 12 2018

STATUS

approved

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Last modified March 20 13:18 EDT 2019. Contains 321345 sequences. (Running on oeis4.)