A057331 a(n) = smallest prime p such that the first n iterates of p under x->2x+1 are all primes.

2, 2, 2, 2, 2, 89, 1122659, 19099919, 85864769, 26089808579, 554688278429, 554688278429, 4090932431513069, 95405042230542329

Offset: 0

Author

Patrick De Geest, Aug 15 2000

Keywords

• nonn
• nice
• more

Comments

For n > 10, a(n) == -1 (mod 2*3*5*11*13). - Farideh Firoozbakht, Apr 24 2004

From Glen Whitney, Sep 14 2022: (Start)

Extending Firoozbakht's observation, modulo any prime p, all residues of a(n) of the form 2^k - 1 mod p are forbidden for n greater than or equal to the number of such residues, e.g., a(n) may not be congruent to 1 or 3 mod 7 for n >= 2.

A067849(a(n)) >= n and for each odd a(n) that occurs in this sequence, (a(n)-1)/2 occurs in A321058. (End)

Examples

			a(5) = 89 because the numbers 89, 179, 359, 719, 1439, 2879 are all primes and 89 is the first number to have this property.
		

Links

• C. K. Caldwell, Latest results about Cunningham Chains
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

See A067849 (number of prime iterates starting from any n) and A321058 (starting points that yield record numbers of iterates).

See also A005602.

Programs

Mathematica

f[n_] := 2n + 1; k = 1; Do[ While[ Union[ PrimeQ[ NestList[ f, Prime[k], n]]] != {True}, k++ ]; Print[ Prime[k]], {n, 1, 9} ]

PARI

has(p,n)=for(k=1,n,if(!isprime(p), return(0)); p=2*p+1); isprime(p)
a(n)=forprime(p=2,, if(has(p,n), return(p))) \\ Charles R Greathouse IV, Apr 29 2015

Extensions

More terms from Farideh Firoozbakht, Apr 24 2004

a(11) (from the Caldwell link) sent by Peter Deleu, Hulste, Belgium, Nov 22 2004

a(13) added from A005602, Paul Zimmermann, Mar 09 2018