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Search: id:A061712
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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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+0
11
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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, 8581545983, 16911433727, 32212254719, 51539607551, 206141652991, 266287972351, 1030791102463, 824633720831, 2196875771903, 4260607557631, 15384572854271, 17591112302591, 26388279066623, 70368744046591, 140735340871679, 246290604621823, 560750930165759, 1125762467889151, 2251799813685119, 4362862139015167, 9005000231485439, 17996806323437567, 63050394783186911, 72053195991416831, 108086391056891903, 504403149675560959, 576460752169205759, 1152917106560335871, 2305834213120671743
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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 (maxal(AT)cs.ucsd.edu), Aug 03 2005
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1024
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FORMULA
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Conjecture: a(n) < 2^(n+3). - T. D. Noe, Mar 14 2008
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EXAMPLE
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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
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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
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Do[k = 1; While[ Count[ IntegerDigits[ Prime[k], 2], 1] != n, k++ ]; Print[ Prime[k]], {n, 1, 30} ]
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CROSSREFS
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Cf. A001348.
Cf. A000043, A000668, A110699, A110700.
Sequence in context: A093363 A127581 A118883 this_sequence A059661 A072686 A002230
Adjacent sequences: A061709 A061710 A061711 this_sequence A061713 A061714 A061715
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KEYWORD
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nonn,nice
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AUTHOR
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Alex Healy (ahealy(AT)fas.harvard.edu), Jun 19 2001
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EXTENSIONS
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Extended to 60 terms by Max Alekseyev (maxal(AT)cs.ucsd.edu), Aug 03 2005
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