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A225876 Composite n which divide s(n)+1, where s is the linear recurrence sequence s(n) = -s(n-1) + s(n-2) - s(n-3) + s(n-5) with initial terms (5, -1, 3, -7, 11). 1
4, 14791044, 143014853, 253149265, 490434564, 600606332, 993861182, 3279563483 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The pseudoprimes derived from the fifth-order 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 pre-tests 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

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Last modified December 9 10:32 EST 2016. Contains 278971 sequences.