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A235463 Primes whose base-6 representation also is the base-2 representation of a prime. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

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.

MATHEMATICA

  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 *)

PROG

(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)

CROSSREFS

Cf. A065720A036952, A065721 - A065727, A235394, A235395, A089971A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Sequence in context: A106924 A076285 A077720 * A127313 A217561 A003521

Adjacent sequences:  A235460 A235461 A235462 * A235464 A235465 A235466

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 11 2014

STATUS

approved

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Last modified September 17 09:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)