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Primes whose base-4 representation is also the base-6 representation of a prime.
2

%I #17 Jan 16 2022 23:22:03

%S 2,3,5,13,17,37,61,73,109,157,173,181,229,233,241,257,317,337,349,373,

%T 397,409,541,557,569,601,613,661,761,769,797,821,857,953,1013,1021,

%U 1033,1069,1153,1181,1193,1201,1229,1237,1297,1321,1373,1429,1481,1609,1621,1637,1709,1801,1861,1877,1889,1901,1973

%N Primes whose base-4 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="/A235624/b235624.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 5 = 11_4 and 11_6 = 7 are both prime, so 5 is a term.

%t Select[Prime@ Range@ 500, PrimeQ@ FromDigits[ IntegerDigits[#, 4], 6] &] (* _Giovanni Resta_, Sep 12 2019 *)

%o (PARI) is(p,b=6,c=4)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ Note: This code is only valid for b > c.

%Y Cf. A235616, 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