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 A061712 Smallest prime with Hamming weight n (i.e., with exactly n 1's when written in binary). 23
 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; text; internal format)
 OFFSET 1,1 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, Aug 03 2005 Drmota, Mauduit, & Rivat prove that a(n) exists for n > N for some N. - Charles R Greathouse IV, May 17 2010 A000120(a(n)) = A014499(A049084(a(n))) = n. - Reinhard Zumkeller, Feb 10 2013 LINKS T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..3320 (first 1024 terms from T. D. Noe) Michael Drmota, Christian Mauduit, and Joel Rivat, Primes with an average sum of digits, Compositio Mathematica 145 (2009), pp. 271-292. MathOverflow, Are there primes of every Hamming weight? 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. FORMULA Conjecture: a(n) < 2^(n+3). - T. D. Noe, Mar 14 2008 EXAMPLE The fourth term is 23 (10111 in binary), since no prime less than 23 has exactly 4 1's in its binary representation. 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); # Max Alekseyev, Aug 03 2005 MATHEMATICA Do[k = 1; While[ Count[ IntegerDigits[ Prime[k], 2], 1] != n, k++ ]; Print[ Prime[k]], {n, 1, 30} ] (* Second program: *) a[n_] := Module[{m, s, k, p}, For[m=0, True, m++, s = {1, Sequence @@ #, 1} & /@ Permutations[Join[Table[1, {n-2}], Table[0, {m}]]] // Sort; For[k=1, k <= Length[ s], k++, p = FromDigits[s[[k]], 2]; If[PrimeQ[p], Print["a(", n, ") = ", p]; Return[p]]]]]; a = 2; Array[a, 100] (* Jean-François Alcover, Mar 16 2015 *) Module[{hw=Table[{n, DigitCount[n, 2, 1]}, {n, Prime[Range[250*10^6]]}]}, Table[ SelectFirst[hw, #[]==k&], {k, 31}]][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 01 2019 *) PROG (Haskell) a061712 n = fromJust \$ find ((== n) . a000120) a000040_list -- Reinhard Zumkeller, Feb 10 2013 (PARI) a(n)=forprime(p=2, , if (hammingweight(p) == n, return(p)); ); \\ Michel Marcus, Mar 16 2015 CROSSREFS Cf. A000043, A000668, A001348, A066195, A110699, A110700. Sequence in context: A127581 A278477 A118883 * A059661 A214704 A231075 Adjacent sequences:  A061709 A061710 A061711 * A061713 A061714 A061715 KEYWORD nonn,base,nice AUTHOR Alex Healy, Jun 19 2001 EXTENSIONS Extended to 60 terms by Max Alekseyev, Aug 03 2005 Further terms from T. D. Noe, Mar 14 2008 STATUS approved

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Last modified June 26 10:12 EDT 2019. Contains 324375 sequences. (Running on oeis4.)