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A235265 Primes whose base 3 representation also is the base 2 representation of a prime. 66
3, 13, 31, 37, 271, 283, 733, 757, 769, 1009, 1093, 2281, 2467, 2521, 2551, 2917, 3001, 3037, 3163, 3169, 3187, 3271, 6673, 7321, 7573, 9001, 9103, 9733, 19801, 19963, 20011, 20443, 20521, 20533, 20749, 21871, 21961, 22123, 22639, 22717, 27253, 28711, 28759, 29173, 29191, 59077, 61483, 61507, 61561, 65701, 65881 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This sequence and A235383 and A229037 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014

This sequence was motivated by work initiated by V.J. Pohjola's post to the SeqFan list, which led to a clarification of the definition and correction of some errors, in sequences A089971, A089981 and A090707 through A090721. These sequences use "rebasing" (terminology of A065361) from some base b to base 10. Sequences A065720 - A065727 follow the same idea but use rebasing in the other sense, from base 10 to base b. The observation that only (10,b) and (b,10) had been considered so far led to the definition of this and related sequences: In a systematic approach, it seems natural to start with the smallest possible pairs of different bases, (2,3) and (3,2), then (2 <-> 4), (3 <-> 4), (2 <-> 5), etc.

Among the two possibilities using the smallest possible bases, 2 and 3, the present one seems a little bit more interesting, among others because not every base-3 representation is a valid base-2 representation (in contrast to the opposite case). This is also a reason why the present sequence grows much faster than the partner sequence A235266.

LINKS

Robert Israel, 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

Veikko Pohjola, A090709 Decimal primes whose decimal representation in base 6 is also prime., SeqFan list, Jan 03 2014

EXAMPLE

E.g., 3 = 10[3] and 10[2] = 2 is prime. 13 = 111[3] and 111[2] = 7 is prime.

MAPLE

N:= 1000: # to get the first N terms

count:= 0:

for i from 1 while count < N do

   p2:= ithprime(i);

   L:= convert(p2, base, 2);

   p3:= add(3^(j-1)*L[j], j=1..nops(L));

   if isprime(p3) then

      count:= count+1;

      A235265[count]:= p3;

   fi

od:

[seq(A235265[i], i=1..N)]; # Robert Israel, May 04 2014

MATHEMATICA

b32pQ[n_]:=Module[{idn3=IntegerDigits[n, 3]}, Max[idn3]<2&&PrimeQ[ FromDigits[ idn3, 2]]]; Select[Prime[Range[7000]], b32pQ] (* Harvey P. Dale, Apr 24 2015 *)

PROG

(PARI) is(p, b=2, c=3)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)

CROSSREFS

Cf. A235266, A065720A036952, A065721 - A065727, A235394, A235395, A089971A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Sequence in context: A273769 A097955 A077717 * A275081 A248368 A171517

Adjacent sequences:  A235262 A235263 A235264 * A235266 A235267 A235268

KEYWORD

nonn,base,nice

AUTHOR

M. F. Hasler, Jan 05 2014

STATUS

approved

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Last modified June 25 11:48 EDT 2017. Contains 288709 sequences.