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A235354
Minimal k > 1 such that the base-k representation of the n-th prime, read in decimal, is also prime.
11
3, 2, 2, 4, 4, 4, 4, 4, 2, 4, 7, 4, 5, 4, 2, 4, 7, 4, 3, 4, 4, 3, 4, 2, 4, 2, 3, 4, 4, 4, 6, 4, 8, 3, 2, 4, 2, 2, 4, 2, 2, 3, 4, 3, 4, 2, 3, 8, 4, 2, 4, 7, 4, 4, 8, 10, 10, 9, 3, 5, 3, 4, 3, 4, 2, 4, 2, 6, 10, 3, 7, 4, 2, 3, 2, 2, 4, 10, 4, 3, 4, 3, 10, 3, 3
OFFSET
1,1
COMMENTS
Conjecture 1. Every number 2, ..., 10 occurs infinitely many times.
Conjecture 2. There exists limit of average (a(1) + ... + a(n))/n.
Conjecture: The average in Conjecture 2 exists and is equal to 10. - Charles R Greathouse IV, Jan 08 2014
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
Prime(7) = 17. The base 2 representation of 17 is 10001, which reinterpreted in decimal is 73 * 137; the base 3 representation of 17 is 122, which reread as decimal is 2 * 61; and the base 4 representation of 17 is 101, which reread as decimal is prime, so therefore a(7) = 4.
PROG
(PARI) rebase(n, from, to=10)=subst(Pol(digits(n, from)), 'x, to)
a(n)=my(p=prime(n)); for(b=2, 9, if(isprime(rebase(p, b)), return(b))); 10 \\ Charles R Greathouse IV, Jan 08 2014
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Vladimir Shevelev, Jan 07 2014
EXTENSIONS
More terms from Peter J. C. Moses
STATUS
approved