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

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

Author

Vladimir Shevelev, Nov 08 2010

Keywords

• nonn

Comments

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).

Links

• 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.

Crossrefs

Cf. A181595, A181692, A181701, A000043.

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

Programs

Haskell

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

Maple

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

Mathematica

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

PARI

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

Python

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

Formula

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

Extensions

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