A095078 -id:A095078 - cp's OEIS frontend

A095058 Number of primes with a single 0-bit (A095078) in range ]2^n,2^(n+1)].

0, 1, 2, 2, 3, 0, 4, 4, 3, 1, 5, 1, 4, 0, 3, 2, 8, 1, 11, 4, 5, 0, 7, 1, 2, 0, 1, 5, 4, 0, 7, 5, 1, 1, 9, 0, 6, 0, 7, 1, 6, 0, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 0, 4, 3, 0, 1, 8, 3, 3, 0, 3, 1, 8, 1, 2, 2, 3, 0, 9, 1, 5, 2, 5, 8, 3, 0, 10, 3, 0, 2, 4, 4, 6, 1

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

For large n, the average value of a(n) is about 4. See A138290 for the n such that a(n)=0. - T. D. Noe, Mar 14 2008

T. D. Noe, Table of n, a(n) for n=1..2048
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 3. - From _N. J. A. Sloane_, Jun 19 2011.
Samuel S. Wagstaff, Jr., Prime Numbers with a fixed number of one bits or zero bits in their binary representation, Exp. Math. vol. 10, issue 2 (2001) 267, Table 3.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A095018.

a(n) = sum(k=2^n+1, 2^(n+1), isprime(k) && (#select(x->x==0, binary(k))==1)); \\ Michel Marcus, Sep 11 2015

A100724 Prime numbers whose binary representations are split into at most 3 runs.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 47, 59, 61, 67, 71, 79, 97, 103, 113, 127, 131, 191, 193, 199, 223, 227, 239, 241, 251, 257, 263, 271, 383, 449, 463, 479, 487, 499, 503, 509, 769, 911, 967, 991, 1009, 1019, 1021, 1031, 1039, 1087, 1151, 1279, 1543, 1567

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 11 2004

The n-th prime is a term iff A100714(n) <= 3.

			a(3)=5 is a term because it is the 3rd prime whose binary representation splits into no more than 3 runs: 5_10 = 101_2.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Run-Length Encoding.

Includes A000668 and A095078.

Cf. A100714, A000040.

R:= 2,3: count:= 2:
for d from 2 while count < 100 do
  for a from d-1 to 1 by -1 do
    for b from 0 to a-1 do
      p:= 2*(2^d - 2^a + 2^b)-1;
      if isprime(p) then R:= R,p; count:= count+1 fi
  od od;
  p:= 2^(d+1)-1;
  if isprime(p) then R:= R,p; count:= count+1 fi
od:
R; # Robert Israel, Oct 30 2024

Select[Table[Prime[k], {k, 1, 50000}], Length[Split[IntegerDigits[ #, 2]]] <= 3 &]

A117890 Numbers k such that number of non-leading 0's in binary representation of k divides k.

Original entry on oeis.org

2, 4, 5, 6, 10, 11, 12, 13, 14, 16, 18, 22, 23, 24, 26, 27, 28, 29, 30, 36, 40, 42, 46, 47, 48, 54, 55, 58, 59, 60, 61, 62, 65, 75, 76, 78, 80, 84, 88, 90, 94, 95, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 118, 119, 120, 122, 123, 124, 125, 126, 132, 140, 144, 145

Offset: 1

Leroy Quet, Mar 30 2006

Contains primes of A095078(n) as a subset. Intersection of a(n) with A049445(n) is A117891(n). - R. J. Mathar, Apr 03 2006

			24 is 11000 in binary. This binary representation has three 0's and 3 divides 24. So 24 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A049445, A117891.

Cf. A023416.

#include 
int main(int argc, char *argv[]) { for(int n=1; n< 500; n++) { int digs=0; int nshifted=n; while(nshifted) { digs += 1- nshifted & 1; nshifted >>= 1; } if ( digs) if( n % digs == 0 ) printf("%d, ",n); } } // R. J. Mathar, Apr 03 2006

a117890 n = a117890_list !! (n-1)
a117890_list = [x | x <- [1..], let z = a023416 x, z > 0, mod x z == 0]
-- Reinhard Zumkeller, Mar 31 2015

a(n) <= A117891(n). - R. J. Mathar, Apr 03 2006

a(n) mod A023416(a(n)) = 0. - Reinhard Zumkeller, Nov 22 2007

More terms from R. J. Mathar, Apr 03 2006

A272143 For a given n, and any m less than n-1, the total number of primes of the form 2^n-2^m-1.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 0, 4, 4, 3, 1, 5, 1, 4, 0, 3, 2, 8, 1, 11, 4, 5, 0, 7, 1, 2, 0, 1, 5, 4, 0, 7, 5, 1, 1, 9, 0, 6, 0, 7, 1, 6, 0, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 0, 4, 3, 0, 1, 8, 3, 3, 0, 3, 1, 8, 1, 2, 2, 3, 0, 9, 1, 5, 2, 5, 8, 3, 0, 10

Offset: 1

Hans Havermann, Apr 21 2016

nonn

For the first 12000 terms the average is ~3.8 with a maximum of 25 at a(11520).

Essentially the same as A095058. - R. J. Mathar, Apr 24 2016

			For n=1, m<0, so there are no solutions. For n=2 there is one solution: m=0, yielding prime 2. For n=3, one solution: m=1, yielding prime 5. For n=4 there are two solutions: m=2 and m=1, yielding primes 11 and 13 respectively. The primes so formed are terms of A095078.

Hans Havermann, Table of n, a(n) for n = 1..12000
Hans Havermann, Table of n, {m1, m2, ...} for primes of the form 2^n-2^m-1, m

Cf. A095078.

Table[Length[Select[Table[2^n - 2^m - 1, {m, 0, n - 2}], PrimeQ[#] & ]], {n, 1, 100}] (* Robert Price, Apr 21 2016 *)

from sympy import isprime
def a(n): return sum(1 for i in range(n-1) if isprime(2**n-1-2**i))
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 09 2023

A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

Original entry on oeis.org

2, 2039, 6143, 522239, 33546239, 260046847, 16911433727, 32212254719, 2196875771903, 140735340871679, 2251799813685119, 9005000231485439, 576460752169205759, 36893488147410714623, 147573811852188057599, 9444732965739282038783, 154742504910672534362390399

Offset: 1

Ya-Ping Lu, Dec 23 2023

It seems that most of the terms end with '9', followed by those ending with '3', '7', and '1'.

			2 is a term because 2 is a prime with one '0' in binary form ('10') and '01' is not a prime. 2039 is a term because 2039 is a prime with one '0' in binary form ('11111110111') and changing the position of the '0', for example, '11111111011' = 2043 and '01111111111' = 1023, always results in a composite.

Subsequence of A095078.

Cf. A039986, A081118, A095058, A138290, A208083.

from sympy import isprime
for n in range(1,100):
    s = n*'1'; c = 0
    for j in range(n+1):
        num = int(s[:j]+'0'+s[j:], 2)
        if isprime(num):
            c += 1
            if c == 1: r = num
            if c == 2: break
    if c == 1: print(r, end = ', ')

cp's OEIS Frontend

A095058 Number of primes with a single 0-bit (A095078) in range ]2^n,2^(n+1)].

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A100724 Prime numbers whose binary representations are split into at most 3 runs.

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A117890 Numbers k such that number of non-leading 0's in binary representation of k divides k.

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A272143 For a given n, and any m less than n-1, the total number of primes of the form 2^n-2^m-1.

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Mathematica

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A368214 Primes with a single 0-bit in binary expansion such that changing the position of the 0-bit always gives a nonprime (including the one with a leading zero).

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Python