A087522 -id:A087522 - cp's OEIS frontend

A127582 a(n) = the smallest prime number of the form k*2^n - 1, for k >= 1.

2, 3, 3, 7, 31, 31, 127, 127, 1279, 3583, 5119, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

			a(0)=2 because 2 = 3*2^0 - 1 is prime.
a(1)=3 because 3 = 2*2^1 - 1 is prime.
a(2)=3 because 3 = 1*2^2 - 1 is prime.
a(3)=7 because 7 = 1*2^3 - 1 is prime.
a(4)=31 because 31 = 2*2^4 - 1 is prime.

Robert Israel, Table of n, a(n) for n = 0..3310

Cf. A007522, A127575-A127581, A127583-A127587.

A087522 is identical except for a(1).

p:= 2: A[0]:= 2:
for n from 1 to 100 do
  if p+1 mod 2^n = 0 then A[n]:= p
  else
    p:=p+2^(n-1);
    while not isprime(p) do p:= p+2^n od:
    A[n]:= p;
  fi
od:
seq(A[i],i=0..100); # Robert Israel, Jan 13 2017

a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 0, 50}]; a (* Artur Jasinski, Jan 19 2007 *)

a(n) << 37^n by Xylouris's improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Edited by Don Reble, Jun 11 2007

Further edited by N. J. A. Sloane, Jul 03 2008

A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.

Original entry on oeis.org

2, 1, 0, 2, 3, 2, 4, 4, 3, 10, 3, 3, 2, 7, 2, 25, 6, 17, 4, 13, 3, 20, 36, 20, 11, 27, 66, 23, 39, 24, 19, 13, 3, 10, 6, 122, 71, 58, 24, 13, 3, 2, 41, 10, 6, 32, 58, 17, 4, 79, 26, 55, 36, 48, 31, 28, 9, 2, 76, 24, 32, 28, 63, 20, 37, 9, 2, 7, 39, 10, 91, 47

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

Michael S. Branicky, Table of n, a(n) for n = 0..2391

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593, A127594, A127598.

a = {}; Do[k = 0; While[ !PrimeQ[k 4^n + (4^n - 1)/3], k++ ]; AppendTo[a, k], {n, 0, 50}]; a (*Artur Jasinski*)
lnk[n_]:=Module[{k=0,n4=4^n},While[!PrimeQ[k*n4+(n4-1)/3],k++];k]; Array[ lnk,60,0] (* Harvey P. Dale, May 28 2018 *)

from sympy import isprime
def a(n):
    k, fourn = 0, 4**n
    while not isprime(k*fourn + (fourn-1)//3): k += 1
    return k
print([a(n) for n in range(72)]) # Michael S. Branicky, May 18 2022

Offset corrected and a(51) and beyond from Michael S. Branicky, May 18 2022

A127598 Least primes of the form k 4^n + (4^n-1)/3.

Original entry on oeis.org

2, 5, 5, 149, 853, 2389, 17749, 70997, 218453, 2708821, 3495253, 13981013, 39146837, 492131669, 626349397, 27201459541, 27201459541, 297784399189, 297784399189, 3665038759253, 3665038759253, 89426945725781

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589, A127590, A127591, A127592, A127593, A127594, A127597.

a = {}; Do[k = 0; While[ !PrimeQ[k 4^n + (4^n - 1)/3], k++ ]; AppendTo[a, k 4^n + (4^n - 1)/3], {n, 0, 50}]; a (*Artur Jasinski*)

A307503 Least prime containing at least n consecutive 1's in its binary representation.

Original entry on oeis.org

2, 2, 3, 7, 31, 31, 127, 127, 1021, 3583, 4093, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 4194301, 14680063, 16777213, 67108859, 536870909, 536870909, 536870909, 536870909, 2147483647, 2147483647, 2147483647, 2147483647, 21474836479

Offset: 0

John Mason, Apr 11 2019

For n > 0, a(n) = A000040(m) for the lowest m such that A090000(m) >= n.

a(n) = A087522(n) for n = 0 through 7, and in all other cases when a(n) is a base 2 repunit (Mersenne) prime.

			a(0) = 2, the smallest prime containing >= 0 1's.
a(1) = 2, the smallest prime containing >= 1 consecutive 1's.
a(2) = 3, the smallest prime containing >= 2 consecutive 1's.

Chai Wah Wu, Table of n, a(n) for n = 0..3314

Cf. A038374, A090000, A087522, A201914.

Cf. A090593 (with exactly n consecutive ones).

nbo(n)=if (n==0, return (0)); n>>=valuation(n, 2); if(n<2, return(n)); my(e=valuation(n+1, 2)); max(e, nbo(n>>e)); \\ A038374
a(n) = my(p=2); while(nbo(p) < n, p=nextprime(p+1)); p; \\ Michel Marcus, Apr 14 2019

a(n) <= A201914(n). - Rémy Sigrist, Apr 11 2019

a(n) = min_{k>=n} A090593(k). - Chai Wah Wu, Apr 26 2019

a(28)-a(32) from Chai Wah Wu, Apr 26 2019

cp's OEIS Frontend

A127582 a(n) = the smallest prime number of the form k*2^n - 1, for k >= 1.

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A127597 Least number k such that k 4^n + (4^n-1)/3 is prime.

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A127598 Least primes of the form k 4^n + (4^n-1)/3.

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Author

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Programs

Mathematica

A307503 Least prime containing at least n consecutive 1's in its binary representation.

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