A235354 Minimal k > 1 such that the base-k representation of the n-th prime, read in decimal, is also prime.

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

Cf. A000040, A052033, A235377, A236174, A236437.

Sequence in context: A332502 A287870 A308190 * A058743 A117970 A364846

Adjacent sequences: A235351 A235352 A235353 * A235355 A235356 A235357

Keyword

nonn,base

Author

Vladimir Shevelev, Jan 07 2014

Extensions

More terms from Peter J. C. Moses

Status

approved