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Primes whose base-5 representation is also the base-6 representation of a prime.
2

%I #17 Jan 16 2022 23:30:01

%S 2,3,11,31,71,131,191,211,241,251,271,331,421,431,461,491,541,601,631,

%T 811,821,911,971,1031,1051,1061,1171,1181,1201,1231,1291,1321,1361,

%U 1451,1511,1531,1571,1601,1721,1801,1811,1831,1861,1931,2081,2111,2131,2141,2311,2341,2381,2411,2521,2531,2711,2741,2801

%N Primes whose base-5 representation is also the base-6 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.

%H Giovanni Resta, <a href="/A235625/b235625.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 11 = 21_5 and 21_6 = 13 are both prime, so 11 is a term.

%t Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 5], 6] &] (* _Giovanni Resta_, Sep 12 2019 *)

%o (PARI) is(p,b=6,c=5)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

%Y Cf. A235626, A235265, A235266, A152079, A235461 - A235482, A065720 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235615 - A235639. See the LINK for further cross-references.

%K nonn,base

%O 1,1

%A _M. F. Hasler_, Jan 13 2014