A235467 Primes whose base-4 representation also is the base-3 representation of a prime.

2, 89, 137, 149, 281, 293, 353, 389, 409, 421, 593, 613, 661, 1097, 1109, 1289, 1301, 1321, 1381, 1409, 1601, 1609, 1669, 2069, 2129, 2309, 2377, 2389, 2729, 4133, 4229, 4373, 4441, 4513, 4673, 5153

Offset: 1

Author

M. F. Hasler, Jan 12 2014

Keywords

• nonn
• base

Comments

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

This is a subsequence of A002144, A002313, A003655, A050150, A062090, A141293, A175768, A192592, A226181 (conjectural).

Examples

			E.g., 89 = 1121_4 and 1121_3 = 43 both are prime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Crossrefs

Cf. A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Programs

Mathematica

b4b3Q[n_]:=Module[{b4=IntegerDigits[n,4]},Max[b4]<3&&PrimeQ[ FromDigits[ b4,3]]]; Select[Prime[Range[700]],b4b3Q] (* Harvey P. Dale, Dec 14 2021 *)

PARI

is(p,b=3,c=4)=vecmax(d=digits(p,c))
				

PARI

forprime(p=1,1e3,is(p,4,3)&&print1(vector(#d=digits(p,3),i,4^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,4)