A081091 Primes of the form 2^i + 2^j + 1, i > j > 0.

7, 11, 13, 19, 37, 41, 67, 73, 97, 131, 137, 193, 521, 577, 641, 769, 1033, 1153, 2053, 2081, 2113, 4099, 4129, 8209, 12289, 16417, 18433, 32771, 32801, 32833, 40961, 65539, 133121, 147457, 163841, 262147, 262153, 262657, 270337, 524353, 524801

Offset

1,1

Comments

This is sequence A070739 without the Fermat primes, A000215. Sequence A081504 lists the i for which there are no primes. - T. D. Noe, Jun 22 2007

Primes in A014311. - Reinhard Zumkeller, May 03 2012

Links

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..7800 (n = 1..1000 from T. D. Noe)

Richard Ehrenborg and N. Bradley Fox, The Descent Set Polynomial Revisited, arXiv:1408.6858 [math.CO], 2014.

Norman B. Fox, Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians, University of Kentucky, Theses and Dissertations, Mathematics, Paper 25.

Formula

A000120(a(n)) = 3.

Example

    7 = 2^2 + 2^1 + 1
   11 = 2^3 + 2^1 + 1
   13 = 2^3 + 2^2 + 1
   19 = 2^4 + 2^1 + 1
   37 = 2^5 + 2^2 + 1
   41 = 2^5 + 2^3 + 1
   67 = 2^6 + 2^1 + 1
   73 = 2^6 + 2^3 + 1
   97 = 2^6 + 2^5 + 1
  131 = 2^7 + 2^1 + 1
  137 = 2^7 + 2^3 + 1
  193 = 2^7 + 2^6 + 1
  521 = 2^9 + 2^3 + 1

Maple

N:= 20: # to get all terms < 2^N
select(isprime, [seq(seq(2^i+2^j+1, j=1..i-1), i=1..N-1)]); # Robert Israel, May 17 2016

Mathematica

Select[Flatten[Table[2^i + 2^j + 1, {i, 21}, {j, i-1}]], PrimeQ] (* Alonso del Arte, Jan 11 2011 *)

Prog

(PARI) do(mx)=my(v=List(), t); for(i=2, mx, for(j=1, i-1, if(ispseudoprime(t=2^i+2^j+1), listput(v, t)))); Vec(v) \\ Charles R Greathouse IV, Jan 02 2014
(PARI) is(n)=hammingweight(n)==3 && isprime(n) \\ Charles R Greathouse IV, Aug 28 2017
(PARI) A81091=[7]; next_A081091(p, i=exponent(p), j=exponent(p-2^i))=!until(isprime(2^i+2^j+1), j++>=i && i++ && j=1)+2^i+2^j)
A081091(n)={for(k=#A81091, n-1, A81091=concat(A81091, next_A081091(A81091[k]))); A81091[n]} \\ M. F. Hasler, Mar 03 2023
(Haskell)
a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
-- Reinhard Zumkeller, May 03 2012
(Python)
from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A081091_gen(): # generator of terms
    return filter(isprime, map(lambda s:int('1'+''.join(s)+'1', 2), (s for l in count(1) for s in multiset_permutations('0'*(l-1)+'1'))))
A081091_list = list(islice(A081091_gen(), 30)) # Chai Wah Wu, Jul 19 2022

Crossrefs

Essentially the same as A070739.

Cf. A000040, A000215, A081092, A010051.

Cf. A095077 (primes with four bits set).

A057733 = 2^A057732 + 3 and A039687 = 3*2^A002253 + 1 are subsequences.

Sequence in context: A059308 A075521 A084444 * A027901 A129213 A110966

Adjacent sequences: A081088 A081089 A081090 * A081092 A081093 A081094

Keyword

nonn,easy

Author

Reinhard Zumkeller, Mar 05 2003

Status

approved