A091936 -id:A091936 - cp's OEIS frontend

A092099 Smallest prime between 2^n and 2^(n+1) having a minimal number of 1's in binary representation, A091936(n) - 2^n.

0, 1, 3, 1, 5, 3, 3, 1, 9, 9, 5, 3, 17, 33, 3, 1, 2049, 3, 65, 33, 17, 129, 9, 513, 35, 131073, 32769, 3, 32769, 3, 65, 81, 17, 513, 16385, 8193, 9, 2049, 33554433, 97, 65, 129, 515, 131073, 129, 32769, 5, 21, 1073741825, 8388609, 65, 2097153, 5, 8589934593, 3, 81

Offset: 1

Robert G. Wilson v, Feb 19 2004

Cf. A091936.

(* First run the program for A091936 to define f[n] *) Join[{0}, Table[ f[n] - 2^n, {n, 2, 56}]] (* Robert G. Wilson v *)

A091936(n) - A000079(n).

A091938 Smallest prime between 2^n and 2^(n+1), having a maximal number of 1's in binary representation.

Original entry on oeis.org

3, 7, 11, 31, 47, 127, 191, 383, 991, 2039, 3583, 8191, 15359, 20479, 63487, 131071, 245759, 524287, 786431, 1966079, 4128767, 7323647, 14680063, 33546239, 67108351, 100646911, 260046847, 536739839, 1073479679, 2147483647

Offset: 1

Reinhard Zumkeller, Feb 14 2004

nonn

A091937(n) = A000120(a(n)).

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A091936, A000668.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; Do[c = 0; While[p < 2^n, b = Count[ IntegerDigits[p, 2], 1]; If[c < b, c = b; q = p]; p = NextPrim[p]]; Print[q], {n, 1, 30}] (* Robert G. Wilson v, Feb 21 2004 *)
b[n_] := Min[ Select[ FromDigits[ #, 2] & /@ (Join[{1}, #, {1}] & /@ Permutations[ Join[{0}, Table[1, {n - 2}]]]), PrimeQ[ # ] &]]; c[n_] := Min[ Select[ FromDigits[ #, 2] & /@ (Join[{1}, #, {1}] & /@ Permutations[ Join[{0, 0}, Table[1, {n - 3}]]]), PrimeQ[ # ] &]]; f[n_] := If[ PrimeQ[2^(n + 1) - 1], 2^(n + 1) - 1, If[ PrimeQ[ b[n]], b[n], c[n]]]; Table[ f[n], {n, 30}] (* Robert G. Wilson v *)

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A091938(n):
    for i in range(n,-1,-1):
        q = 2**n
        for d in multiset_permutations('0'*(n-i)+'1'*i):
            p = q+int(''.join(d),2)
            if isprime(p):
                return p # Chai Wah Wu, Apr 08 2020

More terms from Robert G. Wilson v, Feb 20 2004

A091935 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 3

Offset: 1

Reinhard Zumkeller, Feb 14 2004

nonn

a(n) = A000120(A091936(n)).

0 never appears, 1 appears only at 1, 2's appear only for Fermat primes (A019434), 4's appear at A092100. I have found no fives <= 250. - Robert G. Wilson v

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A091937, A092100.

f:= proc(n) local i,j,k;
  if isprime(2^n+1) then return 2 fi;
  for i from 1 to n-1 do if isprime(2^n+1+2^i) then return 3 fi od;
  for i from 1 to n-2 do for j from i+1 to n-1 do if isprime(2^n+2^i+2^j+1) then return 4 fi od od;
  error ">=5 found"
end proc:
f(1):= 1:
map(f, [$1..200]); # Robert Israel, Mar 30 2020

Run the second Mathematica line of A091936, then Join[{1}, Count[ IntegerDigits[ #, 2], 1] & /@ Table[ f[n], {n, 2, 105}]] (* Robert G. Wilson v, Feb 19 2004 *)

More terms from Robert G. Wilson v, Feb 18 2004

A333876 a(n) is the largest prime 2^(n-1) <= p < 2^n minimizing the Hamming weight of all primes in this interval.

Original entry on oeis.org

2, 5, 13, 17, 41, 97, 193, 257, 769, 1153, 2113, 4129, 12289, 18433, 40961, 65537, 163841, 270337, 786433, 1179649, 2101249, 4194433, 8650753, 16777729, 50332673, 69206017, 167772161, 270532609, 537133057, 1107296257, 3221225473, 6442713089, 8858370049

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..1000

Cf. A066195, A091936, A333877, A333878.

for(n=2, 10, my(hmin=n+n,pmax); forprime(p=2^(n-1), 2^n, my(h=hammingweight(p)); if(h<=hmin,pmax=p;hmin=h)); print1(pmax,", "))

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A333876(n):
    for i in range(n):
        q = 2**n-1
        for d in multiset_permutations('0'*i+'1'*(n-1-i)):
            p = q-int(''.join(d),2)
            if isprime(p):
                return p # Chai Wah Wu, Apr 08 2020

A333878 a(n) is the number of primes 2^(n-1) <= p < 2^n with minimal Hamming weight A091935(n-1) in this interval.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 1, 4, 2, 3, 2, 2, 2, 4, 1, 3, 4, 5, 3, 2, 1, 5, 1, 32, 2, 5, 2, 2, 8, 6, 37, 5, 3, 4, 2, 3, 2, 2, 57, 3, 5, 52, 1, 5, 3, 7, 63, 1, 2, 5, 1, 5, 2, 6, 87, 6, 99, 2, 3, 2, 1, 2, 102, 2, 3, 5, 3, 6, 2, 2, 2, 5, 2, 7, 1, 3, 2, 3, 1, 6, 2, 4, 3, 3

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

Giovanni Resta, Table of n, a(n) for n = 2..1000

Cf. A091935, A091936, A333876, A333879.

a[2] = 1; a[n_] := Block[{c = 0, nb = 0}, While[c == 0, c = Count[ 2^(n-1) + 1 + Total /@ Subsets[ 2^Range[n - 2], {nb}], x_ /; PrimeQ[x]]; nb++]; c]; Array[a, 90, 2] (* Giovanni Resta, Apr 09 2020 *)

Terms a(35) and beyond from Giovanni Resta, Apr 09 2020

A092100 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1) is 4.

Original entry on oeis.org

25, 32, 40, 43, 48, 56, 58, 64, 96, 104, 112, 120, 128, 134, 140, 145, 152, 160, 176, 185, 192, 208, 212, 224, 235, 240, 244, 248, 252, 256, 264, 272, 280, 286, 288, 292, 302, 304, 308, 320, 326, 332, 348, 356, 360, 384, 392, 394, 400

Offset: 1

Robert G. Wilson v, Feb 19 2004

nonn

Where 4 appears in A091935.
This sequence differs from multiples of 8 (A008590) very little but significantly; even fewer are odd.
Essentially the same as A081504. - R. J. Mathar, Sep 08 2008

Cf. A091935, A091936.

Compute the second line of the Mathematica code for A091936, then Do[ If[ Count[ IntegerDigits[ f[n], 2], 1] == 4, Print[n]], {n, 1, 400}] (* Robert G. Wilson v, Feb 19 2004 *)

cp's OEIS Frontend

A092099 Smallest prime between 2^n and 2^(n+1) having a minimal number of 1's in binary representation, A091936(n) - 2^n.

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A091938 Smallest prime between 2^n and 2^(n+1), having a maximal number of 1's in binary representation.

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A091935 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1).

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A333876 a(n) is the largest prime 2^(n-1) <= p < 2^n minimizing the Hamming weight of all primes in this interval.

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A333878 a(n) is the number of primes 2^(n-1) <= p < 2^n with minimal Hamming weight A091935(n-1) in this interval.

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Extensions

A092100 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1) is 4.

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Mathematica