A074788 Prime numbers in the Perrin sequence b(n+1) = b(n-1) + b(n-2) with initial values b(1)=3, b(2)=0, b(3)=2.

2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797, 22584751787583336797527561822649328254745329, 29918426252927024136988188355201180399482197

Offset

1,1

Comments

a(18) has 114 digits; a(19) has 128 digits. - Harvey P. Dale, Aug 11 2011

Links

Michael De Vlieger, Table of n, a(n) for n = 1..24

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Eric Weisstein's World of Mathematics, Perrin Prime.

Formula

a(n+1) = a(n-1)+a(n-2) if a(n+1) is prime and a(1) = 3, a(2) = 0, a(3) = 2

Example

a(1)=3, a(2)=0, a(3)=2; then for n = 3, a(4) = a(2) + a(1) = 0 + 3 = 3; for n = 4, a(5) = a(3) + a(2) = 2 + 0 = 2 etc

Mathematica

a[1] = 3; a[2] = 0; a[3] = 2; a[n_] := a[n] = a[n - 2] + a[n - 3]; Do[ If[ PrimeQ[ a[n]], Print[a[n]]], {n, 1, 357}]
(* Alternative: *)
Union[Select[LinearRecurrence[{0, 1, 1}, {3, 0, 2}, 500], PrimeQ]] (* Harvey P. Dale, Aug 11 2011 *)
(* Alternative: *)
Select[RootSum[-1 - # + #^3 &, #^Range[1000] &], PrimeQ] // Union (* Eric W. Weisstein, Jun 05 2026 *)

Prog

(PARI) aprime(n)= a=vector(n+1); a[1]=3; a[2]=0; a[3]=2; print("n a(n+1)"); for(x=3, n, a[x+1]=a[x-1]+a[x-2]; if(isprime(a[x+1]), print("a("x+1") = "a[x+1])) )

Crossrefs

Cf. A112881, A001608.

Sequence in context: A289757 A030480 A048418 * A262833 A070805 A255161

Adjacent sequences: A074785 A074786 A074787 * A074789 A074790 A074791

Keyword

nonn,changed

Author

Cino Hilliard, Sep 07 2002

Extensions

Edited by Robert G. Wilson v, Sep 13 2002

Status

approved