A002256 -id:A002256 - cp's OEIS frontend

A050528 Primes of the form 9*2^n+1.

19, 37, 73, 577, 1153, 18433, 147457, 1179649, 77309411329, 39582418599937, 79164837199873, 83010348331692982273, 332041393326771929089, 1328165573307087716353, 21760664753063325144711169, 196002643346460554954903773880698489798657

Offset: 1

N. J. A. Sloane, Dec 29 1999

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..25
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

For more terms see A002256.

[a: n in [0..150] | IsPrime(a) where a is (9*2^n+1) ]; // Vincenzo Librandi, Aug 09 2014

A050528:=n->`if`(isprime(9*2^n+1),9*2^n+1,NULL): seq(A050528(n), n=1..200); # Wesley Ivan Hurt, Aug 09 2014

Select[9*2^Range[150]+1,PrimeQ] (* Harvey P. Dale, Aug 08 2014 *)

A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

67, 9431, 461081, 2543551

Offset: 1

Arkadiusz Wesolowski, Feb 21 2017

Fernando (Remark 5.2) shows that all terms are odd. - Jeppe Stig Nielsen, Jan 02 2025

R. Fernando, Morehead-like restrictions on Fermat divisors.
Proth Search Page, Prime factors of Fermat numbers
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002256.

Cf. A000215, A050528, A057778, A201364, A204620, A226366, A280003.

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

A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.

Original entry on oeis.org

1, 3, 7, 43, 63, 211

Offset: 1

Ken Clements, Aug 05 2025

The exponent, k, of 2 must be odd because the exponent, 2, of 3 (where 9 = 3^2) is even and the sum of the exponents of prime factors 2 and 3 must be odd to form a product that is a twin prime average. Of all subsequences of A027856, this is the longest known where the power of 3 is fixed.

Amiram Eldar noted that using A002236 and A002256, we obtain a(7) > 5.6*10^6, if it exists.

			a(1) = 1 because 2*9 = 18 with 17 and 19 prime.
a(2) = 3 because 8*9 = 72 with 71 and 73 prime.
a(3) = 7 because 128*9 = 1152 with 1151 and 1153 prime.
a(4) = 43 because 8796093022208*9 = 79164837199872 with 79164837199871 and 79164837199873 prime.

Intersection of A002236 and A002256.

Cf. A014574, A384530, A027856, A385433, A181490.

q:= k-> (m-> andmap(isprime, [m-1, m+1]))(9*2^k):
select(q, [2*i-1$i=1..111])[];  # Alois P. Heinz, Aug 08 2025

from gmpy2 import is_prime
def is_TPpi2(e2, e3):
    N = 2**e2 * 3**e3
    return is_prime(N-1) and is_prime(N+1)
print([k for k in range(1, 100001, 2) if is_TPpi2(k, 2)])

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A050528 Primes of the form 9*2^n+1.

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A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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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.

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A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.

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Maple

Python

cp's OEIS Frontend

A050528 Primes of the form 9*2^n+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A386857 Numbers k such that both 9*2^k - 1 and 9*2^k + 1 are prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Python

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.

A386857 Numbers k such that both 92^k - 1 and 92^k + 1 are prime.