A208083 -id:A208083 - cp's OEIS frontend

A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

1, 3, 5, 7, 11, 13, 15, 23, 27, 29, 31, 47, 55, 59, 61, 63, 95, 111, 119, 123, 125, 127, 191, 223, 239, 247, 251, 253, 255, 383, 447, 479, 495, 503, 507, 509, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043

Offset: 1

Reinhard Zumkeller, Mar 06 2003

T(n,n) = A036563(n+1) = 2^(n+1) - 3.

Numbers of the form 2^t - 2^k - 1, 1 <= k < t.

			Triangle begins:
.......... 1 ......... ................ 1
........ 3...5 ....... .............. 11 101
...... 7..11..13 ..... .......... 111 1011 1101
... 15..23..27..29 ... ...... 1111 10111 11011 11101
. 31..47..55..59..61 . . 11111 101111 110111 111011 111101.

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462, A038463.
Cf. A181741 (primes), A208083, subsequence of A089633.
Cf. A131094.

a081118 n k = a081118_tabl !! (n-1) !! (k-1)
a081118_row n = a081118_tabl !! (n-1)
a081118_tabl  = iterate
   (\row -> (map ((+ 1) . (* 2)) row) ++ [4 * (head row) + 1]) [1]
a081118_list = concat a081118_tabl
-- Reinhard Zumkeller, Feb 23 2012

Table[2^(n+1)-2^(n-k+1)-1,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Apr 09 2020 *)

T(n, k) = 2^(n+1) - 2^(n-k+1) - 1, 1<=k<=n.

a(n) = (2^A002260(n)-1)*2^A004736(n)-1; a(n)=(2^i-1)*2^j-1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

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

A181741 Primes of the form 2^t-2^k-1, k>=1.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 29, 31, 47, 59, 61, 127, 191, 223, 239, 251, 383, 479, 503, 509, 991, 1019, 1021, 2039, 3583, 3967, 4079, 4091, 4093, 6143, 8191, 15359, 16127, 16319, 16381, 63487, 65407, 65519, 129023, 131063, 131071, 245759, 253951, 261631, 261887, 262079, 262111, 262127, 262139

Offset: 1

Vladimir Shevelev, Nov 08 2010

nonn

All Mersenne primes A000668(i) are in the sequence, parametrized by t=A000043(i)+1 and k=A000043(i).
If p is in the sequence, then the exponents t and k are unique.
For given k, the smallest value of t defines sequence A181692.
Every term p=2^t-2^k-1 in this sequence here generates an entry 2^(t-1)*p in A181595 (cf. A181701).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000, probable primes for n > 150
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046. - _N. J. A. Sloane_, Sep 04 2012
Vladimir Shevelev, Perfect and near-perfect numbers, arXiv:1011.6160 [math.NT], 2010-2012.

Cf. A181595, A181692, A181701, A000043.

Cf. A010051, primes in A081118, see also A208083.

a181741 n = a181741_list !! (n-1)
a181741_list = filter ((== 1) . a010051) a081118_list
-- Reinhard Zumkeller, Feb 23 2012

isA000079 := proc(n) if n = 1 then true; elif type(n,'odd') then false; else if nops( numtheory[factorset](n) ) = 1 then  true;  else
false; end if; end if; end proc:
isA181741 := proc(p) if isprime(p) then k := A007814(p+1) ; (p+1)/2^k+1 ; return ( isA000079(%) and k >=1 ) ; else
false;  end if; end proc:
for i from 1 to 1000 do p := ithprime(i) ; if isA181741(p) then printf("%d,",p) ; end if; end do: # R. J. Mathar, Nov 18 2010

Select[Table[2^t-2^k-1, {t, 1, 20}, {k, 1, t-1}] // Flatten // Union, PrimeQ] (* Jean-François Alcover, Nov 16 2017 *)

lista(nn) = {for (n=3, nn, forstep(k=n-1, 1, -1, if (isprime(p=2^n-2^k-1), print1(p, ", "));););} \\ Michel Marcus, Dec 17 2018

from itertools import count, islice
from sympy import isprime
def A181741_gen(): # generator of terms
    m = 2
    for t in count(1):
        r=1<>=1
        m<<=1
A181741_list = list(islice(A181741_gen(),30)) # Chai Wah Wu, Jul 08 2022

Conjecture: equals the intersection of A000040 and A081118 or the intersection of A000040 and A089633. [R. J. Mathar, Nov 18 2010]

Corrected (251 and 1019 inserted) and extended by R. J. Mathar, Nov 18 2010

A208091 Smallest number m such that exactly n primes of the form 2^m - 2^k - 1 exist, 1 <= k < m.

Original entry on oeis.org

1, 11, 3, 4, 6, 8, 38, 24, 32, 18, 48, 138, 20, 588, 144, 252, 5520, 168, 7200, 2400, 2850

Offset: 0

Reinhard Zumkeller, Feb 23 2012

A208083(a(n)) = n and A208083(m) <> n for m < a(n).

a(21) > 7600, if it exists. - Giovanni Resta, Jun 14 2018

			a(3) = 4 because for m = 4 there are exactly three primes of the given form: 13 = 2^4 - 2^1 - 1, 11 = 2^4 - 2^2 - 1, 7 = 2^4 - 2^3 - 1 and no smaller m satisfies this requirement.

Cf. A208083.

import Data.List (elemIndices, elemIndex)
import Data.Maybe (fromJust)
a208091 = (+ 1) . fromJust . (`elemIndex` a208083_list)

f:= n -> nops(select(k -> isprime(2^n-2^k-1), [$1..n-1])):
for n from 1 to 300 do
v:= f(n);
if not assigned(A[v]) then A[v]:= n fi;
od:
seq(A[m],m=0..15); # Robert Israel, Jun 13 2018

A = <||>; Do[c = Length@Select[Range[n-1], PrimeQ[2^n - 2^# - 1] &]; If[! KeyExistsQ[A, c], A[c]=n], {n, 140}]; Array[A, 13, 0] (* Giovanni Resta, Jun 13 2018 *)

a(n) = {my(m=1); while(sum(k=1, m, isprime(2^m-2^k-1)) != n, m++); m;} \\ Michel Marcus, Jun 13 2018

Corrected by Robert Israel, Jun 13 2018
a(17), a(19)-a(20) from Robert Israel, Jun 13 2018
a(16), a(18) from Giovanni Resta, Jun 14 2018

A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

Original entry on oeis.org

1, 7, 15, 23, 27, 31, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 127, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479

Offset: 1

Thomas Ordowski, Jan 22 2024

nonn

Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.
For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.
All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.
Cf. A138290 (see Chai Wah Wu's conjecture in the third comment). By Crocker's (1971) theorem: if m > 2 and a <> b, then 2^(2^m)-1 - 2^a - 2^b is not prime.
If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.
Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.
Prime numbers of this sequence are Mersenne primes > 3, and many other primes.
Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.
If so, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a (dual) Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0.
For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.
It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.
For many terms m, 2m+1 is also in this sequence. By iteration (x -> 2x+1), the subsequence b(n) = (m+1)2^n-1, for n >= 0, is infinite if m = 7 (which has already been proven) and probably if m = 27 (which is hard to prove).

			7 is a term since {2^7-1-2, 2^7-1-2^2, 2^7-1-2^3, 2^7-1-2^4, 2^7-1-2^5, 2^7-1-2^6} = {125, 123, 119, 111, 95, 63} and all six of these numbers are composite.
Note that both 2^148-1 and 2^148+1 are de Polignac numbers.

Cf. A000225, A006285, A000668, A101036, A138290, A208083, A232565, A369378.

fQ[n_] := Block[{k = n -1}, While[k > 1 && !PrimeQ[2^n -1 -2^k], k--]; k == 1]; Select[ Range[3, 450], fQ] (* Robert G. Wilson v, Jan 22 2024 *)

For n > 1, a(n) = A138290(n-1) + 1.

A208083(a(n)) = 0, for n > 0.

More terms from Robert G. Wilson v, Jan 22 2024

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

A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

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

A181741 Primes of the form 2^t-2^k-1, k>=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A208091 Smallest number m such that exactly n primes of the form 2^m - 2^k - 1 exist, 1 <= k < m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Extensions

A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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