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
Keywords
Examples
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.
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.
Programs
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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 *) -
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)
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
Comments