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).

4, 14791044, 143014853, 253149265, 490434564, 600606332, 993861182, 3279563483

Offset: 1

Author

Matt McIrvin, May 23 2013

Keywords

• nonn
• hard
• more

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

Examples

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

Links

• 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)

Programs

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 */

Extensions

Terms 4 through 7 found by Richard Holmes, added by Matt McIrvin, May 27 2013

a(8) from Dana Jacobsen, Aug 29 2016