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A057331
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a(n) = smallest prime p such that the first n iterates of p under x->2x+1 are all primes.
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22
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2, 2, 2, 2, 2, 89, 1122659, 19099919, 85864769, 26089808579, 554688278429, 554688278429, 4090932431513069, 95405042230542329
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OFFSET
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0,1
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COMMENTS
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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)
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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f[n_] := 2n + 1; k = 1; Do[ While[ Union[ PrimeQ[ NestList[ f, Prime[k], n]]] != {True}, k++ ]; Print[ Prime[k]], {n, 1, 9} ]
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PROG
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(PARI) has(p, n)=for(k=1, n, if(!isprime(p), return(0)); p=2*p+1); isprime(p)
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CROSSREFS
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See A067849 (number of prime iterates starting from any n) and A321058 (starting points that yield record numbers of iterates).
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KEYWORD
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nonn,nice,more
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AUTHOR
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EXTENSIONS
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a(11) (from the Caldwell link) sent by Peter Deleu, Hulste, Belgium, Nov 22 2004
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STATUS
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approved
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