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

Author

Vladimir Shevelev, Jan 07 2014

Keywords

• nonn
• base
• changed

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

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

Table[Module[{b=2},While[!PrimeQ[FromDigits[IntegerDigits[p,b]]],b++];b],{p,Prime[Range[90]]}] (* Harvey P. Dale, Aug 30 2025 *)

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

Extensions

More terms from Peter J. C. Moses