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Numbers k such that b(k) is an integer, where b(n) = Sum_{k=0..n} (2*n+k)!/((2*n-2*k)!*(3*k+1)!) (= A001608(6*n+2)/(6*n+2)).

"The column Mathematical Recreations by Ian Stewart in the June 1996 issue of Scientific American discusses the Perrin sequence [A001608] A(n) defined by A(0)=3, A(1)=0, A(2)=2, A(n+1)=A(n-1)+A(n-2). Motivated by a theorem of E. Lucas: If n is prime it divides A(n) exactly, the question whether primality of n follows from n divides A(n) exactly was formulated 1899. So far, they say, nobody has found a composite n that divides A(n). Such a number would be called a Perrin pseudoprime. The article quotes an experiment by Steven Arno of the Supercomputing Research Center in Bowie, Md., where a lower bound of 15 digits for the size of the smallest Perrin pseudoprime was obtained in 1991. On Jul 3rd, 1996, I was able to find the two smallest Perrin pseudoprimes:" [Holzbaur] - Robert G. Wilson v, Nov 30 2001

In the "Feedback" section of his column for November 1996, Ian Stewart mentions that Jeffrey Shallit (Waterloo) had written to him saying that he had found the Perrin pseudoprimes 271441 and 904631 in 1982.

There are 765 Perrin pseudoprimes which are also Carmichael numbers less than 2^64. - Dana Jacobsen, May 10 2015

There are 101994 Perrin pseudoprimes which are also Fermat pseudoprimes to base 2 less than 2^64. - Dana Jacobsen, May 10 2015

Difference in the number of unlabeled maximal independent sets of an a(n)-cycle graph A127687(a(n)) times a(n), from value of Perrin(a(n)) such that Perrin(a(n)) mod a(n) = Sum_{d|a(n)} (Perrin(d)*Phi(a(n)/d)) mod a(n) [dA001608, Phi = A000010. - Richard Turk, Aug 11 2015

There are two known Perrin pseudoprimes that are squares: a(1)=271441=521^2 and a(76)=36366108601=190699^2. In A173656 it is claimed that there are no others < (10^9)^2. - Hugo Pfoertner, Sep 01 2017