A110699 -id:A110699 - cp's OEIS frontend

A061712 Smallest prime with Hamming weight n (i.e., with exactly n 1's when written in binary).

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

Offset: 1

Alexander D. Healy, Jun 19 2001

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

			The fourth term is 23 (10111 in binary), since no prime less than 23 has exactly 4 1's in its binary representation.

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.
Kenichiro Kashihara, Letter to the Editor, Math. Scientist 20 (1) (1995), 67-68.
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.

Cf. A000043, A000120, A000668, A001348, A014499, A049084, A066195, A110699, A110700.

a061712 n = fromJust $ find ((== n) . a000120) a000040_list
-- Reinhard Zumkeller, Feb 10 2013

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

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[1] = 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,#[[2]]==k&],{k,31}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 01 2019 *)

a(n)=forprime(p=2, , if (hammingweight(p) == n, return(p));); \\ Michel Marcus, Mar 16 2015

from itertools import combinations
from sympy import isprime
def A061712(n):
    l, k = n-1, 2**n
    while True:
        for d in combinations(range(l-1,-1,-1),l-n+1):
            m = k-1 - sum(2**(e) for e in d)
            if isprime(m):
                return m
        l += 1
        k *= 2 # Chai Wah Wu, Sep 02 2021

Conjecture: a(n) < 2^(n+3). - T. D. Noe, Mar 14 2008

A000120(a(n)) = A014499(A049084(a(n))) = n. - Reinhard Zumkeller, Feb 10 2013

Extended to 60 terms by Max Alekseyev, Aug 03 2005

Further terms from T. D. Noe, Mar 14 2008

A110700 Number of zeros in the smallest prime with Hamming weight n (given by A061712).

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 0, 1, 1, 1, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Max Alekseyev, Aug 03 2005

a(n)=0 iff n belongs A000043.

Observe that a(n)=3 for n=6, 14, 30, 62, 126, 254, 510, 1022, ... which is A000918. Conjecture: a(n) is never greater than 3. - T. D. Noe, Mar 14 2008

T. D. Noe, Table of n, a(n) for n=1..1024

Cf. A000043, A061712, A110699.

with(combstruct); a:=proc(n) local m,is,s,t,r; if n=1 then return 1 fi; r:=+infinity; for m from 0 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); if isprime(t) then return m fi; od; od; return 0; end;

A061712[n_] := A061712[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], Return[p]]]]]; A061712[1]=2; Table[DigitCount[A061712[n], 2, 0], {n, 1, 100}] (* Jean-François Alcover, Mar 16 2015 *)

a(n) = A110699(n) - n.

A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

Original entry on oeis.org

4, 6, 14, 15, 55, 95, 247, 447, 511, 1535, 2047, 7167, 12287, 32255, 49151, 98303, 196607, 393215, 983039, 1572863, 3145727, 6291455, 8388607, 33423359, 50331647, 117440511, 201326591, 528482303, 805306367, 1879048191, 3221225471

Offset: 1

Jonathan Vos Post, Jun 23 2007

Semiprime analog of A061712. Extended by Stefan Steinerberger. Includes the subset Mersenne semiprimes A092561.

			a(1) = 4 because the first semiprime A001358(1) is 4 (base 10) which is written 100 in binary, the latter representation having exactly 1 one.
a(2) = 6 since A001358(2) = 6 = 110 (base 2) has exactly 2 ones.
a(4) = 15 since A001358(6) = 15 = 1111 (base 2) has exactly 4 ones and, as it also has no zeros, is the smallest of the Mersenne semiprimes.

Donovan Johnson, Table of n, a(n) for n = 1..250

Cf. A000043, A000120, A000337, A000668, A001358, A007088, A061712, A085724, A089226, A089998, A089999, A091991, A092558, A092559, A092561, A092562, A081093, A102782, A110472, A110699, A110700.

Join[{4},Table[SelectFirst[Sort[FromDigits[#,2]&/@Permutations[ Join[ PadRight[{}, n,1],{0}]]],PrimeOmega[#]==2&],{n,2,40}]] (* Harvey P. Dale, Feb 06 2015 *)

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A061712 Smallest prime with Hamming weight n (i.e., with exactly n 1's when written in binary).

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A110700 Number of zeros in the smallest prime with Hamming weight n (given by A061712).

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A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

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