A081504 -id:A081504 - cp's OEIS frontend

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

Reinhard Zumkeller, Mar 05 2003

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

			    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

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.

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.

a081091 n = a081091_list !! (n-1)
a081091_list = filter ((== 1) . a010051') a014311_list
-- Reinhard Zumkeller, May 03 2012

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

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

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

is(n)=hammingweight(n)==3 && isprime(n) \\ Charles R Greathouse IV, Aug 28 2017

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

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

A000120(a(n)) = 3.

A095056 Number of primes with three 1-bits (A081091) in range [2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 3, 0, 4, 2, 3, 2, 2, 2, 4, 1, 3, 4, 5, 3, 2, 1, 5, 1, 0, 2, 5, 2, 2, 8, 6, 0, 5, 3, 4, 2, 3, 2, 2, 0, 3, 5, 0, 1, 5, 3, 7, 0, 1, 2, 5, 1, 5, 2, 6, 0, 6, 0, 2, 3, 2, 1, 2, 0, 2, 3, 5, 3, 6, 2, 2, 2, 5, 2, 7, 1, 3, 2, 3, 1, 6, 2, 4, 3, 3, 2, 6, 1, 1, 5, 7, 2, 4, 2, 5, 0, 3, 4, 3, 1, 2, 1, 3, 0, 5

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

			From _Michael De Vlieger_, Feb 27 2017: (Start)
a(1) = 0 because there are no primes with three 1s in binary expansion between 2^1 and 2^2.
a(2) = 1 since the only prime between 2^2 and 2^3 with three 1s in binary expansion is 7 = binary 111.
a(3) = 2 since between 2^3 and 2^4 we have 11 and 13 (binary 1011 and 1101, respectively) have three 1s.
(End)

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
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 2. - 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. A081504, A095018, A095057.

Table[m = Count[Prime@ Range[PrimePi[2^n] + 1, PrimePi[2^(n + 1) - 1]], k_ /; DigitCount[k, 2, 1] == 3]; Print@ m; m, {n, 24}] (* Michael De Vlieger, Feb 27 2017 *)

More terms from T. D. Noe, Oct 17 2007

A133830 Least positive number k < n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 0, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 0, 17, 15, 1, 15, 1, 6, 0, 4, 9, 14, 13, 3, 11, 25, 0, 6, 7, 0, 17, 7, 15, 2, 0, 30, 23, 6, 21, 2, 33, 1, 0, 3, 0, 14, 5, 6, 21, 19, 0, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27, 33, 4, 3, 26, 1, 39, 35, 19, 9, 18

Offset: 2

T. D. Noe, Sep 26 2007

nonn

Sequence A081504 gives the n such that a(n) = 0. For those n, A133831(n) gives the least k > n for which the binary trinomial is prime.

T. D. Noe, Table of n, a(n) for n=2..2000

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).

Cf. A095056, A133831, A133832 (k > n equivalent).

Mathematica
```
Table[s=1+2^n; k=1; While[k
				
```

Edited by Peter Munn, Sep 30 2024

A092100 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1) is 4.

Original entry on oeis.org

25, 32, 40, 43, 48, 56, 58, 64, 96, 104, 112, 120, 128, 134, 140, 145, 152, 160, 176, 185, 192, 208, 212, 224, 235, 240, 244, 248, 252, 256, 264, 272, 280, 286, 288, 292, 302, 304, 308, 320, 326, 332, 348, 356, 360, 384, 392, 394, 400

Offset: 1

Robert G. Wilson v, Feb 19 2004

nonn

Where 4 appears in A091935.
This sequence differs from multiples of 8 (A008590) very little but significantly; even fewer are odd.
Essentially the same as A081504. - R. J. Mathar, Sep 08 2008

Cf. A091935, A091936.

Compute the second line of the Mathematica code for A091936, then Do[ If[ Count[ IntegerDigits[ f[n], 2], 1] == 4, Print[n]], {n, 1, 400}] (* Robert G. Wilson v, Feb 19 2004 *)

A346146 Numbers m such that there are no primes of the form 2^m + 2^k - 1, for 0 < k < m.

Original entry on oeis.org

1, 9, 17, 25, 29, 33, 43, 45, 49, 53, 57, 59, 69, 73, 81, 89, 97, 103, 113, 129, 134, 143, 161, 165, 173, 193, 201, 205, 206, 209, 225, 227, 229, 233, 237, 241, 257, 273, 278, 281, 289, 293, 297, 303, 305, 321, 345, 349, 353, 369, 377, 381, 383, 385, 401, 405

Offset: 1

Lamine Ngom, Jul 06 2021

nonn

In comparison with A081504 (dealing with 2^m + 2^k + 1) where most of the terms are even, here the vast majority of terms are odd.

Cf. A081504.

q[m_] := AllTrue[Range[m - 1], ! PrimeQ[2^m + 2^# - 1] &]; Select[Range[400], q] (* Amiram Eldar, Jul 06 2021 *)

isok(m) = for(k=1, m-1, if (ispseudoprime(2^m+2^k-1), return (0))); return (1); \\ Michel Marcus, Jul 06 2021

cp's OEIS Frontend

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

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A095056 Number of primes with three 1-bits (A081091) in range [2^n,2^(n+1)].

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A133830 Least positive number k < n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

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A092100 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1) is 4.

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A346146 Numbers m such that there are no primes of the form 2^m + 2^k - 1, for 0 < k < m.

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PARI