A247981 Primes dividing nonzero terms in A003095: the iterates of x^2 + 1 starting at 0.

2, 5, 13, 41, 137, 149, 229, 293, 397, 509, 661, 677, 709, 761, 809, 877, 881, 1217, 1249, 1277, 1601, 2053, 2633, 3637, 3701, 4481, 4729, 5101, 5449, 5749, 5861, 7121, 7237, 7517, 8009, 8089, 8117, 8377, 9661, 14869, 14897, 18229, 19609, 20369, 20441, 21493, 22349, 23917, 24781, 24977, 25717

Offset

1,1

Comments

Relative density in the primes is 0, see Jones theorem 5.5.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Rafe Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2) (2008), pp. 523-544.

Formula

a(n) << exp(k^n) for some constant k > 0, see Jones theorem 6.1. In particular this sequence is infinite. - Charles R Greathouse IV, Sep 28 2014

Example

2 and 13 are in the sequence since A003095(4) = 26. 3 is not in the sequence since it does not divide any member of A003095.

Mathematica

Select[Table[d=0; t=0; Do[t=Mod[t^2+1, Prime[j]]; If[t==0, d=1], {k, 1, Prime[j]}]; If[d==1, Prime[j], 0], {j, 1, 1000}], #!=0&] (* Vaclav Kotesovec, Oct 04 2014 *)

Prog

(PARI) is(p)=my(v=List([1]), t=1); while(t, t=(t^2+1)%p; for(i=1, #v, if(v[i]==t, return(0))); listput(v, t)); isprime(p)

Crossrefs

Cf. A003095, A248218, A248219.

Sequence in context: A263308 A288388 A339224 * A149868 A343175 A007269

Adjacent sequences: A247978 A247979 A247980 * A247982 A247983 A247984

Keyword

nonn

Author

Charles R Greathouse IV, Sep 28 2014

Status

approved