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%I #10 Apr 14 2021 01:17:11
%S 7,2801,17207,19559,134513,134807,840743,842759,842801,941249,943601,
%T 958007,958049,958343,5899657,6591089,6607903,6706393,6722857,
%U 41196751,41311663,41314057,46137673,46137967,46253257,46942351,46944409,47059657
%N Primes whose base-7 representation also is the base-2 representation of a prime.
%C 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.
%C For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
%C When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=7, thus a subsequence of A077721.
%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 = 10_7 and 10_2 = 2 are both prime, so 7 is a term.
%e 2801 = 11111_7 and 11111_2 = 31 are both prime, so 2801 is a term.
%o (PARI) is(p,b=2,c=7)=vecmax(d=digits(p,c))<b&&isprime(vector(#d,i,b^(#d-i))*d~)&&isprime(p)
%o (PARI) forprime(p=1,1e3,is(p,7,2)&&print1(vector(#d=digits(p,2),i,7^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,7)
%Y Cf. A235477, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.
%K nonn,base
%O 1,1
%A _M. F. Hasler_, Jan 11 2014