A225984 -id:A225984 - cp's OEIS frontend

A290139 a(n) = A225984(n) mod n.

0, 1, 2, 3, 4, 3, 6, 3, 2, 8, 10, 5, 12, 3, 8, 3, 16, 6, 18, 6, 0, 14, 22, 9, 9, 16, 11, 11, 28, 18, 30, 3, 26, 3, 33, 2, 36, 3, 6, 34, 40, 26, 42, 0, 2, 26, 46, 17, 41, 43, 27, 24, 52, 42, 28, 43, 12, 32, 58, 26, 60, 3, 60, 3, 23, 11, 66, 45, 62, 17, 70, 42, 72, 3

Offset: 1

Seiichi Manyama, Jul 21 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A225876, A225984.

MapIndexed[Mod[#1, First@ #2] &, LinearRecurrence[{-1, 1, -1, 0, 1}, {-1, 3, -7, 11, -16}, 74]] (* Michael De Vlieger, Jul 21 2017, after T. D. Noe at A225984 *)

a(p) = p - 1 for all prime p.

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).

Original entry on oeis.org

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

Offset: 1

Matt McIrvin, May 23 2013

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

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

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)

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 */

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

a(8) from Dana Jacobsen, Aug 29 2016

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A290139 a(n) = A225984(n) mod n.

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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).

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