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A155967
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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.
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0
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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
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1,1
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COMMENTS
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Note that composites can also be self-dual under this bitwise transpose transformation; i.e., 283 (base 10) = 100011011 (base 2) whose row-by-row 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
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LINKS
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EXAMPLE
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a(1) = 11 because 11 (base 2) = 1011. Write the matrix whose row-by-row is [10,11], read by columns to get 1101 and since 1101 (base 2) = 13, which is prime. Note that such numbers are either self-dual or have a distinct dual, so a(2) = 13. a(3) = 257 because 257 (base 2) = 100000001, whose bit-transpose is itself (a self-dual 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).
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MAPLE
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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 l-1 do for c from 0 to l-1 do b2 := subsop(1+r+l*c=op(1+c+l*r, b), b2) ; od: od: add(op(i, b2)*2^(i-1), 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
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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