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Primes whose base-4 representation is also the base-9 representation of a prime.
3

%I #15 Jan 16 2022 23:30:16

%S 2,3,29,41,61,89,109,149,157,281,293,313,401,421,433,593,701,709,1013,

%T 1049,1061,1069,1097,1117,1277,1289,1301,1553,1601,1709,2069,2137,

%U 2237,2309,2377,2437,2477,2689,2729,2749,2797,2957,2969,3001,3061,3109,3169,3329,3361,3389,3457,3533,3701

%N Primes whose base-4 representation is also the base-9 representation of a prime.

%C This sequence is part of a two-dimensional array of sequences, given in the LINK, 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.

%C Appears to be a subsequence of A197636.

%H Giovanni Resta, <a href="/A235481/b235481.txt">Table of n, a(n) for n = 1..10000</a>

%H M. F. Hasler, <a href="https://docs.google.com/document/d/10IM7fcAbB2tqRGuwfGvuEGUzD_IXbgXPDK0tfxN4M3o/pub">Primes whose base c expansion is also the base b expansion of a prime</a>

%e 29 = 131_4 and 131_9 = 109 are both prime, so 29 is a term.

%t Select[Prime@Range@600, PrimeQ[FromDigits[IntegerDigits[#, 4], 9]] &] (* _Giovanni Resta_, Sep 12 2019 *)

%o (PARI) is(p,b=9,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: Code only valid for b > c.

%Y Cf. A235473 - A235480, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

%K nonn,base

%O 1,1

%A _M. F. Hasler_, Jan 12 2014