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A235463
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Primes whose base-6 representation also is the base-2 representation of a prime.
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2
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7, 37, 43, 223, 1297, 1303, 1549, 7993, 9109, 46663, 54469, 55987, 326593, 1679659, 1681129, 1727569, 1734049, 1967587, 2006461, 2007763, 2014027, 2015287, 10077919, 10125649, 10125691, 10133467, 10412107, 10413397, 11757349, 11766421, 11766427, 11766637
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OFFSET
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1,1
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COMMENTS
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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.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
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=6, thus a subsequence of A077720.
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LINKS
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EXAMPLE
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7 = 11_6 and 11_2 = 3 are both prime, so 7 is a term.
37 = 101_6 and 101_2 = 5 are both prime, so 37 is a term.
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MATHEMATICA
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b62Q[n_]:=Module[{idn6=IntegerDigits[n, 6]}, Max[idn6]<2&&AllTrue[ {FromDigits[ idn6, 6], FromDigits[idn6, 2]}, PrimeQ]]; Select[Prime[ Range[ 4, 780000]], b62Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2020 *)
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PROG
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(PARI) is(p, b=2, c=6)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)
(PARI) forprime(p=1, 1e3, is(p, 6, 2)&&print1(vector(#d=digits(p, 2), i, 6^(#d-i))*d~, ", ")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(., 2, 6)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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