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Primes whose base-4 representation is also the base-8 representation of a prime.
2

%I #17 Jan 16 2022 23:27:00

%S 2,3,7,11,19,29,37,59,71,89,97,107,149,167,223,227,251,281,337,347,

%T 439,461,463,491,499,509,521,563,599,617,647,653,659,701,727,733,739,

%U 751,757,769,797,809,823,877,887,907,911,929,937,1031,1087,1093,1109,1163,1187,1193,1297

%N Primes whose base-4 representation is also the base-8 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="/A235633/b235633.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 7 = 13_4 and 13_8 = 11 are both prime, so 7 is a term.

%t Select[Prime@Range@500, PrimeQ@ FromDigits[IntegerDigits[#, 4], 8] &] (* _Giovanni Resta_, Sep 12 2019 *)

%o (PARI) is(p,b=8,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. A235618, 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