A091938 -id:A091938 - cp's OEIS frontend

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

2, 5, 11, 17, 37, 67, 131, 257, 521, 1033, 2053, 4099, 8209, 16417, 32771, 65537, 133121, 262147, 524353, 1048609, 2097169, 4194433, 8388617, 16777729, 33554467, 67239937, 134250497, 268435459, 536903681, 1073741827, 2147483713

Offset: 1

Reinhard Zumkeller, Feb 14 2004

nonn

A091935(n) = A000120(a(n)).

So far only a(25) and a(32) possess 4 1's in their binary representation.

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

Cf. A091938, A019434.

NextPrim[ n_] := Block[ {k = n + 1}, While[ !PrimeQ[ k], k++ ]; k]; p = 2; Do[ c = Infinity; While[ p < 2^n, b = Count[ IntegerDigits[ p, 2], 1]; If[ c > b, c = b; q = p]; p = NextPrim[ p]; If[ c < 4, p = NextPrim[ 2^n]; Continue[ ]]]; Print[ q], {n, 2, 32}] (* Robert G. Wilson v, Feb 18 2004 *)
b[ n_ ] := Min[ Select[ FromDigits[ #, 2 ] & /@ (Join[ {1}, #, {1} ] & /@ Permutations[ Join[ {1}, Table[ 0, {n - 2} ] ] ]), PrimeQ[ # ] & ] ]; c[ n_ ] := Min[ Select[ FromDigits[ #, 2 ] & /@ (Join[ {1}, #, {1} ] & /@ Permutations[ Join[ {1, 1}, Table[ 0, {n - 3} ] ] ]), PrimeQ[ # ] & ] ]; f[ n_ ] := If[ PrimeQ[ 2^n + 1 ], 2^n + 1, If[ PrimeQ[ b[ n ] ], b[ n ], c[ n ] ] ]; Table[ f[ n ], {n, 2, 32} ] (* Robert G. Wilson v *)

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A091936(n):
    for i in range(n+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 18 2004

A091937 Greatest number of 1's in binary representations of primes between 2^n and 2^(n+1).

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 7, 8, 9, 10, 11, 13, 13, 13, 15, 17, 17, 19, 19, 20, 21, 21, 23, 24, 25, 25, 27, 28, 29, 31, 31, 32, 33, 34, 35, 35, 37, 37, 39, 40, 41, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 55, 56, 56, 58, 59, 61, 61, 61, 63, 64, 65, 66, 67, 68, 69, 69, 71, 72

Offset: 1

Reinhard Zumkeller, Feb 14 2004

nonn

a(n) = A000120(A091938(n)).

Cf. A091935.

Run the second Mathematica line of A091938, then Count[ IntegerDigits[ #, 2], 1] & /@ Table[ f[n], {n, 75}]

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

A333877 a(n) is the largest prime 2^(n-1) <= p < 2^n maximizing the Hamming weight of all primes in this interval.

Original entry on oeis.org

3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, 65519, 131071, 262139, 524287, 1048573, 2097143, 4194301, 8388587, 16777213, 33546239, 67108859, 134217467, 260046847, 536870909, 1073741567, 2147483647, 4294967291, 8589934583, 16911433727

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

This differs from A014234 at n=1 and then first at n=16: a(16) = 65519 != 65521 = A014234(16). - Alois P. Heinz, Apr 25 2020

Chai Wah Wu, Table of n, a(n) for n = 2..998

Cf. A014234, A091937, A091938, A333876, A333879.

a:= proc(n) option remember; local i, p;
      for i from 0 do p:= max(select(isprime, map(l-> add(l[j]*
        2^(j-1), j=1..n), combinat[permute]([1$(n-i),0$i]))));
        if p>0 then break fi
      od; p
    end:
seq(a(n), n=2..30);  # Alois P. Heinz, Apr 23 2020

a[n_] := a[n] = MaximalBy[{#, DigitCount[#, 2, 1]}& /@ Select[Range[ 2^(n-1), 2^n-1], PrimeQ], Last][[-1, 1]];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 30}] (* Jean-François Alcover, Nov 09 2020 *)

for(n=2, 30, my(hmax=0,pmax); forprime(p=2^(n-1), 2^n, my(h=hammingweight(p)); if(h>=hmax,pmax=p;hmax=h)); print1(pmax,", "))

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A333877(n):
    for i in range(n-1,-1,-1):
        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

A333879 a(n) is the number of primes 2^(n-1) <= p < 2^n with maximal Hamming weight A091937(n-1) in this interval.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 4, 3, 1, 5, 1, 4, 18, 3, 1, 8, 1, 11, 4, 5, 31, 7, 1, 2, 33, 1, 5, 4, 1, 7, 5, 1, 1, 9, 55, 6, 71, 7, 1, 6, 69, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 95, 4, 3, 61, 1, 8, 1, 3, 110, 3, 1, 8, 1, 2, 2, 3, 98, 9, 1, 5, 2, 5, 8, 3, 125, 10, 3, 81, 2

Offset: 2

Hugo Pfoertner, Apr 08 2020

nonn

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

Cf. A091937, A091938, A333877, A333878.

a[2]=1; a[n_] := Block[{c = 0, nb = 0}, While[c == 0, c = Count[2^n - 1 - Total /@ Subsets[2^Range[n - 2], {nb}], x_ /; PrimeQ[x]]; nb++]; c]; a /@ Range[2, 85] (* Giovanni Resta, Apr 09 2020 *)

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

A092111 a(n) = n+1 minus the greatest number of 1's in the binary representations of primes between 2^n and 2^(n+1).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 20 2004

nonn

0's occur only at Mersenne prime exponents (A000043) - 1, twos are in A092112, threes do not appear < 504.

a(n) <= 2 for n <= 2000. - Robert Israel, Mar 05 2020

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

Cf. A091938, A092112.

f:= proc(n) local t,j,k;
  t:= 2^(n+1)-1;
  if isprime(t) then return 0 fi;
  for j from 1 to n-1 do if isprime(t-2^j) then return 1 fi od;
  for j from 1 to n-2 do for k from j+1 to n-1 do
    if isprime(t-2^j-2^k) then return 2 fi od od;
  FAIL
end proc:
map(f, [$1..200]); # Robert Israel, Mar 05 2020

Compute the second line of the Mathematica code for A091938, then (Table[n + 1, {n, 105}]) - (Count[ IntegerDigits[ #, 2], 1] & /@ Table[ f[n], {n, 105}])

a(n) = n+1 - A091937(n).

A092112 Where A092111 equals 2.

Original entry on oeis.org

14, 22, 26, 36, 38, 42, 54, 57, 62, 70, 78, 81, 90, 94, 110, 122, 132, 134, 138, 142, 147, 150, 158, 166, 168, 171, 172, 174, 178, 182, 190, 194, 198, 206, 210, 222, 238, 254, 285, 294, 312, 315, 318, 334, 336, 350, 366, 372, 382, 405, 414, 416, 432, 434, 446, 454

Offset: 1

Robert G. Wilson v, Feb 20 2004, corrected Nov 02 2006

nonn

Not as obvious as A092100, this sequence differs from multiples of 8 plus 6 (A017137).

Cf. A092111, A091935.

Run the second Mathematica line of A091938, then g[n_] := (n + 1 - Count[ IntegerDigits[f[n], 2], 1]); Select[ Range[100], g[ # ] == 2 &]

cp's OEIS Frontend

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

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A091937 Greatest number of 1's in binary representations of primes between 2^n and 2^(n+1).

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A333877 a(n) is the largest prime 2^(n-1) <= p < 2^n maximizing the Hamming weight of all primes in this interval.

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A333879 a(n) is the number of primes 2^(n-1) <= p < 2^n with maximal Hamming weight A091937(n-1) in this interval.

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A092111 a(n) = n+1 minus the greatest number of 1's in the binary representations of primes between 2^n and 2^(n+1).

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A092112 Where A092111 equals 2.

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