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A061712
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Smallest prime with Hamming weight n (i.e. with exactly n 1's when written in binary).
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17
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2, 3, 7, 23, 31, 311, 127, 383, 991, 2039, 3583, 6143, 8191, 73727, 63487, 129023, 131071, 522239, 524287, 1966079, 4128767, 16250879, 14680063, 33546239, 67108351, 201064447, 260046847, 536739839, 1073479679, 5335154687, 2147483647
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| a(n) = 2^n - 1 for n in A000043, so Mersenne primes A000668 is a subsequence of this one. Binary length of a(n) is given by A110699 and the number of zeros in a(n) is given by A110700. - Max Alekseyev (maxale(AT)gmail.com), Aug 03 2005
Drmota, Mauduit, & Rivat prove that a(n) exists for n > N. [From Charles R Greathouse IV (charles.greathouse(AT)case.edu), May 11 2010; corrected May 17 2010]
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LINKS
| Charles R Greathouse IV, Table of n, a(n) for n = 1..3320
Michael Drmota, Christian Mauduit, and Joel Rivat, Primes with an average sum of digits, Compositio Mathematica 145 (2009), pp. 271-292. [From Charles R Greathouse IV (charles.greathouse(AT)case.edu), May 11 2010]
MathOverflow, Are there primes of every Hamming weight? [From T. D. Noe (noe(AT)sspectra.com), May 14 2010]
Samuel S. Wagstaff, Prime numbers with a fixed number of one bits or zero bits in their binary representation, Experimental Mathematics 10 (2001), pp. 267-273. [From Charles R Greathouse IV (charles.greathouse(AT)case.edu), May 11 2010]
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FORMULA
| Conjecture: a(n) < 2^(n+3). - T. D. Noe, Mar 14 2008
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EXAMPLE
| The fourth term is 23 (10111 in binary), since no prime less than 23 has exactly 4 1's in its binary representation.
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MAPLE
| with(combstruct); a:=proc(n) local m, is, s, t, r; if n=1 then return 2 fi; r:=+infinity; for m from 0 to 100 do is := iterstructs(Combination(n-2+m), size=n-2); while not finished(is) do s := nextstruct(is); t := 2^(n-1+m)+1+add(2^i, i=s); # print(s, t); if isprime(t) then r:=min(t, r) fi; od; if r<+infinity then return r fi; od; return 0; end; seq(a(n), n=1..60); [Alekseyev]
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MATHEMATICA
| Do[k = 1; While[ Count[ IntegerDigits[ Prime[k], 2], 1] != n, k++ ]; Print[ Prime[k]], {n, 1, 30} ]
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CROSSREFS
| Cf. A001348.
Cf. A000043, A000668, A110699, A110700.
Sequence in context: A093363 A127581 A118883 * A059661 A072686 A002230
Adjacent sequences: A061709 A061710 A061711 * A061713 A061714 A061715
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KEYWORD
| nonn,nice
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AUTHOR
| Alex Healy (ahealy(AT)post.harvard.edu), Jun 19 2001
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EXTENSIONS
| Extended to 60 terms by Max Alekseyev (maxale(AT)gmail.com), Aug 03 2005
Further terms from T. D. Noe, Mar 14 2008
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