A095058 -id:A095058 - cp's OEIS frontend

A095078 Primes with a single 0 bit in their binary expansion.

2, 5, 11, 13, 23, 29, 47, 59, 61, 191, 223, 239, 251, 383, 479, 503, 509, 991, 1019, 1021, 2039, 3583, 3967, 4079, 4091, 4093, 6143, 15359, 16127, 16319, 16381, 63487, 65407, 65519, 129023, 131063, 245759, 253951, 261631, 261887, 262079

Offset: 1

Antti Karttunen, Jun 01 2004

Except for the first value 2, the sequence gives the primes of the form 2^k -2^j -1 with 0 < j < k-1. If j=k-1 we obtain the Mersenne primes. - Pierre CAMI, May 19 2005

{2} UNION (A000040 INTERSECT A190620). - Chai Wah Wu, Dec 19 2024

T. D. Noe, Table of n, a(n) for n=1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A030130. Cf. A095058, A190620.

Select[Prime[Range[23000]],DigitCount[#,2,0]==1&] (* Harvey P. Dale, Nov 28 2019 *)

forprime(p=2,262079,v=binary(p);s=0;for(k=1,#v,s+=v[k]);if(#v-s==1,print1(p,", "))) \\ Washington Bomfim, Jan 13 2011

A138290 Numbers m such that 2^(m+1) - 2^k - 1 is composite for all 0 <= k < m.

Original entry on oeis.org

6, 14, 22, 26, 30, 36, 38, 42, 54, 57, 62, 70, 78, 81, 90, 94, 110, 122, 126, 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

Offset: 1

T. D. Noe, Mar 13 2008

nonn

The binary representation of 2^(m+1) - 2^k - 1 has m 1-bits and one 0-bit. Note that prime m are very rare: 577 is the first and 5569 is the second.
A208083(a(n)+1) = 0 (cf. A081118). - Reinhard Zumkeller, Feb 23 2012 [Corrected by Thomas Ordowski, Feb 19 2024]
Conjecture: 2^j - 2 are terms for j > 2. - Chai Wah Wu, Sep 07 2021
The proof of this conjecture is in A369375. - Thomas Ordowski, Mar 20 2024

			6 is here because 95, 111, 119, 123, 125 and 126 are all composite.

Chai Wah Wu, Table of n, a(n) for n = 1..996 (terms 1..275 from T. D. Noe)

Many common terms with A092112.

Cf. A081118, A095058, A110700, A208083, A369375.

import Data.List (elemIndices)
a138290 n = a138290_list !! (n-1)
a138290_list = map (+ 1) $ tail $ elemIndices 0 a208083_list
-- Reinhard Zumkeller, Feb 23 2012

t={}; Do[num=2^(n+1)-1; k=0; While[kHarvey P. Dale, Apr 09 2022 *)

isok(m) = my(nb=0); for (k=0, m-1, if (!ispseudoprime(2^(m+1) - 2^k - 1), nb++, break)); nb==m; \\ Michel Marcus, Sep 13 2021

from sympy import isprime
A138290_list = []
for n in range(1,10**3):
    k2, n2 = 1, 2**(n+1)
    for k in range(n):
        if isprime(n2-k2-1):
                break
        k2 *= 2
    else:
        A138290_list.append(n) # Chai Wah Wu, Sep 07 2021

For these m, A095058(m) = 0 and A110700(m) > 1.

For n > 0, a(n) = A369375(n+1) - 1. - Thomas Ordowski, Mar 20 2024

A095059 Number of primes with two 0-bits (A095079) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 0, 9, 5, 14, 4, 16, 9, 18, 0, 21, 21, 21, 7, 41, 22, 31, 5, 37, 20, 33, 14, 37, 45, 47, 0, 69, 31, 36, 34, 55, 34, 71, 10, 60, 50, 69, 22, 81, 52, 59, 5, 97, 71, 79, 42, 67, 86, 95, 13, 103, 61, 81, 47, 98, 50, 110, 0, 108, 87, 116, 36, 125, 98, 98, 29, 126, 90, 125, 46, 107, 100, 125, 8, 158, 81, 109, 65, 156, 94, 131, 27, 127, 144, 146, 38, 167, 129, 137, 6, 127, 112, 178, 76

Offset: 1

Antti Karttunen, Jun 01 2004

Michael S. Branicky, Table of n, a(n) for n = 1..1000
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 4. - From _N. J. A. Sloane_, Jun 19 2011.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A095018, A095056, A095057, A095058.

from sympy import isprime
from itertools import combinations, count, islice
def a(n): # generator of terms
    if n < 2: return 0
    b, d = (1<Michael S. Branicky, Dec 27 2023

Added terms a(34)-a(99) from the Wagstaff paper. - N. J. A. Sloane, Jun 19 2011

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

A095078 Primes with a single 0 bit in their binary expansion.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A138290 Numbers m such that 2^(m+1) - 2^k - 1 is composite for all 0 <= k < m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A095059 Number of primes with two 0-bits (A095079) in range ]2^n,2^(n+1)].

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python