

A155967


Binary transpose primes. Integers of k^2 bits which, when written row by row as a square matrix and then read column by column, are primes once transformed.


0



11, 13, 257, 271, 277, 283, 293, 307, 317, 331, 337, 353, 359, 367, 383, 389, 409, 431, 433, 443, 449, 461, 463, 467, 479, 491, 503, 509, 32797, 32801, 32831, 32869, 32887, 32911, 32969, 32987, 32999, 33029, 33049, 33083, 33091, 33161, 33181, 33191
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OFFSET

1,1


COMMENTS

Note that composites can also be selfdual under this bitwise transpose transformation; i.e., 283 (base 10) = 100011011 (base 2) whose rowbyrow matrix is [100,011,011] which is invariant under the transpose. Hence the basic sequence can be called "primes which are fixed points under binary matrix transpose." What are some nontrivial solutions base 10? Base k for other k?
The primes which stay fixed under the transpositions are 257, 283, 433, 443, 32801, 33029, 33377, 33623, 33637, 33811, 34369, 34679, ...  R. J. Mathar, Feb 06 2009


LINKS

Table of n, a(n) for n=1..44.


EXAMPLE

a(1) = 11 because 11 (base 2) = 1011. Write the matrix whose rowbyrow is [10,11], read by columns to get 1101 and since 1101 (base 2) = 13, which is prime. Note that such numbers are either selfdual or have a distinct dual, so a(2) = 13. a(3) = 257 because 257 (base 2) = 100000001, whose bittranspose is itself (a selfdual binary prime). a(4) = 271 because 271 (base 2) = 100001111, which is by rows [100,001,111], which when read by columns is 101001011 and that (base 2) is 331, a prime and the dual, equal to a(9).


MAPLE

A070939 := proc(n) max(1, ilog2(n)+1) ; end: bintr := proc(n) local b, l, b2, r, c ; b := convert(n, base, 2) ; l := sqrt(nops(b)) ; b2 := [seq(0, i=1..l^2)] ; for r from 0 to l1 do for c from 0 to l1 do b2 := subsop(1+r+l*c=op(1+c+l*r, b), b2) ; od: od: add(op(i, b2)*2^(i1), i=1..l^2) ; end: for n from 1 to 4000 do p := ithprime(n) ; if issqr(A070939(p)) then tr := bintr(p) ; if isprime(tr) then printf("%d, ", p) ; fi; fi; od: # R. J. Mathar, Feb 06 2009


CROSSREFS

Cf. A000040, A004676.
Sequence in context: A238090 A093605 A288304 * A111070 A110115 A073765
Adjacent sequences: A155964 A155965 A155966 * A155968 A155969 A155970


KEYWORD

base,easy,nonn


AUTHOR

Jonathan Vos Post, Jan 31 2009


EXTENSIONS

More terms from R. J. Mathar, Feb 06 2009


STATUS

approved



