A002254 -id:A002254 - cp's OEIS frontend

A282946 Numbers k such that 5*2^k + 1 is a prime factor of a generalized Fermat number 11^(2^m) + 1 for some m.

15, 1947, 125413, 240937

Offset: 1

Arkadiusz Wesolowski, Feb 25 2017

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
OEIS Wiki, Generalized Fermat numbers

Cf. A199592, A226366, A268661, A268662, A268663, A282945, A268664.

Subsequence of A002254.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[11, p]], AppendTo[lst, n]], {n, 1, 1947, 2}]; lst

A322301 Primes p such that 5*2^p + 1 is also prime.

Original entry on oeis.org

3, 7, 13, 127, 3313, 23473, 819739

Offset: 1

Vincenzo Librandi, Dec 19 2018

Primes in A002254.

Cf. A002254, A175172.

[p: p in PrimesUpTo (10000) | IsPrime(5*2^p+1)];

select(p->isprime(p) and isprime(5*2^p+1),[$0..5000]); # Muniru A Asiru, Dec 19 2018

Select[Prime[Range[4000]], PrimeQ[5 2^# + 1] &]

A381815 Smallest k>1 such that 10k^(32^n)+1 is prime.

Original entry on oeis.org

3, 2, 2, 2, 138, 24, 695, 107, 250, 404, 4657, 2185, 27931

Offset: 0

Jakub Buczak, Mar 07 2025

			a(0) = 3, because 10*3^(3*2^0)+1 equals 271 which is prime.
a(1) = 2, because 10*2^(3*2^1)+1 equals 641 which is prime.

Cf. A007283, A002254, A089319.

Cf. A381793.

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(2) if isprime(k**(3*(2**n)) * 10 + 1))

a(10)-a(11) from Michael S. Branicky, Mar 07 2025

a(12) from Georg Grasegger, Apr 15 2025

A346542 Numbers k such that 5*2^k + 1 is an elite prime (A102742).

Original entry on oeis.org

3, 15, 55, 26607, 209787, 819739

Offset: 1

Arkadiusz Wesolowski, Sep 16 2021

An integer k is in this sequence if and only if there is no solution to the congruence x^2 == 2^(2^k) + 1 (mod p), where p is a prime of the form 5*2^k + 1.

a(7) > 9*10^6.

Alexander Aigner, Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind, Monatsh. Math., Vol. 101 (1986), pp. 85-93; alternative link.

Subsequence of A002254.

Cf. A102742.

isok(k)=my(p=5*2^k+1); k>2 && Mod(k, 2)==1 && Mod(3, p)^((p-1)/2)+1==0 && kronecker(lift(Mod(2, p)^2^k)+1, p)==-1;

A361076 Array, read by ascending antidiagonals, whose n-th row consists of the powers of 2, if n = 1; of the primes of the form (2n-1)2^k+1, if they exist and n > 1; and of zeros otherwise.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 3, 5, 8, 1, 4, 7, 6, 16, 1, 2, 6, 13, 8, 32, 2, 3, 3, 14, 15, 12, 64, 1, 8, 5, 6, 20, 25, 18, 128, 3, 2, 10, 7, 7, 26, 39, 30, 256, 6, 15, 4, 20, 19, 11, 50, 55, 36, 512, 1, 10, 27, 9, 28, 21, 14, 52, 75, 41, 1024, 1, 4, 46, 51, 10, 82, 43, 17, 92, 85, 66, 2048

Offset: 1

Lorenzo Sauras Altuzarra, Mar 01 2023

Is a(n) <= A279709(n)?

			Table starts
  1   2   4   8  16  32  64 128 ... A000079
  1   2   5   6   8  12  18  30 ... A002253
  1   3   7  13  15  25  39  55 ... A002254
  2   4   6  14  20  26  50  52 ... A032353
  1   2   3   6   7  11  14  17 ... A002256
  1   3   5   7  19  21  43  81 ... A002261
  2   8  10  20  28  82 188 308 ... A032356
  1   2   4   9  10  12  27  37 ... A002258
  ...
(2*39279 - 1)*2^r + 1 is composite for every r > 0 (see comments from A046067), so the 39279th row is A000004, the zero sequence.

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300.

Cf. A000079, A002253, A002254, A032353, A002256, A002261, A032356, A002258.
Cf. A033809 (1st column).
Cf. A000004, A046067, A279709.

vk(k, nn) = if (k==1, return (vector(nn, i, 2^(i-1)))); my(v = vector(nn-k+1), nb=0, i=0, x); while (nb != nn-k+1, if (isprime((2*k-1)*2^i+1), nb++; v[nb] = i); i++;); v;
lista(nn) = my(v=vector(nn, k, vk(k, nn))); my(w=List()); for (i=1, nn, for (j=1, i, listput(w, v[i-j+1][j]););); Vec(w); \\ Michel Marcus, Mar 03 2023

A377248 Numbers k such that 8191 * 2^k + 1 is prime.

Original entry on oeis.org

12, 20, 412, 712, 2092, 4704, 10176, 33396, 41124, 105604, 139780, 142924

Offset: 1

Arsen Vardanyan, Oct 21 2024

8191 is the 5th Mersenne prime: 8191 = 2^13 - 1 (a term of A000668).

			12 is a term, because 8191 * 2^12 + 1 = 8191 * 4096 + 1 = 33550337 is prime. (also a term of A061644).

Cf. A000668, A002253, A002254, A001771, A061644.

PARI
```
is(k) = isprime(8191 * 2^k + 1);
```

a(8)-a(9) from Hugo Pfoertner, Oct 21 2024

a(10)-a(12) from Michael S. Branicky, Nov 05 2024

cp's OEIS Frontend

A282946 Numbers k such that 5*2^k + 1 is a prime factor of a generalized Fermat number 11^(2^m) + 1 for some m.

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A322301 Primes p such that 5*2^p + 1 is also prime.

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A381815 Smallest k>1 such that 10*k^(3*2^n)+1 is prime.

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A346542 Numbers k such that 5*2^k + 1 is an elite prime (A102742).

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PARI

A361076 Array, read by ascending antidiagonals, whose n-th row consists of the powers of 2, if n = 1; of the primes of the form (2*n-1)*2^k+1, if they exist and n > 1; and of zeros otherwise.

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PARI

A377248 Numbers k such that 8191 * 2^k + 1 is prime.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A381815 Smallest k>1 such that 10k^(32^n)+1 is prime.

A361076 Array, read by ascending antidiagonals, whose n-th row consists of the powers of 2, if n = 1; of the primes of the form (2n-1)2^k+1, if they exist and n > 1; and of zeros otherwise.