

A225876


Composite n which divide s(n)+1, where s is the linear recurrence sequence s(n) = s(n1) + s(n2)  s(n3) + s(n5) with initial terms (5, 1, 3, 7, 11).


1




OFFSET

1,1


COMMENTS

The pseudoprimes derived from the fifthorder linear recurrence A225984(n) are analogous to the Perrin pseudoprimes A013998, and the Lucas pseudoprimes A005845.
For prime p, A225984(p) == p  1 (mod p). The pseudoprimes are composite numbers satisfying the same relation. 4 = 2^2; 14791044 = 2^2 * 3 * 19 * 29 * 2237; 143014853 = 907 * 157679.
Like the Perrin test, the modular sequence is periodic so simple pretests can be performed. Numbers divisible by 2, 3, 4, 5, 9, and 25 have periods 31, 11, 62, 24, 33, and 120 respectively.  Dana Jacobsen, Aug 29 2016
a(9) > 1.4*10^11.  Dana Jacobsen, Aug 29 2016


LINKS

Table of n, a(n) for n=1..8.
K. Brown, Proof of Generalized Little Theorem of Fermat, proves that for prime p, a(p) == a(1) (mod p) for recurrences of the form of A225984.
R. Holmes, comments to M. McIrvin's post on Google+ (found terms 4 through 7)


EXAMPLE

A225984(4) = 11, and 11 == 3 (mod 4). Since 4 is composite, it is a pseudoprime with respect to A225984.


PROG

(PARI)
N=10^10;
default(primelimit, N);
M = [0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 1, 0, 1, 1, 1];
a(n)=lift( trace( Mod(M, n)^n ) );
ta(n)=lift( trace( Mod(M, n) ) );
{ for (n=2, N,
if ( isprime(n), next() );
if ( a(n)==ta(n), print1(n, ", "); );
); }
/* Matt McIrvin, after Joerg Arndt's program for A013998, May 23 2013 */


CROSSREFS

Sequence in context: A204041 A065248 A116141 * A067508 A034250 A058436
Adjacent sequences: A225873 A225874 A225875 * A225877 A225878 A225879


KEYWORD

nonn,hard,more


AUTHOR

Matt McIrvin, May 23 2013


EXTENSIONS

Terms 4 through 7 found by Richard Holmes, added by Matt McIrvin, May 27 2013
a(8) from Dana Jacobsen, Aug 29 2016


STATUS

approved



