A002253 -id:A002253 - cp's OEIS frontend

A326655 Numbers k such that 3*4^k+1 is prime.

1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, 1104, 1408, 1584, 1956, 17175, 21147, 24075, 27396, 27591, 40095, 354984, 400989, 916248, 1145805, 2541153, 5414673

Offset: 1

Richard N. Smith, Jul 16 2019

One half of the even terms in A002253.

Numbers k such that r*(r+1)^k+1 is prime: A003306 (r=2), this sequence (r=3), A204322 (r=4), A247260 (r=5), A245241 (r=6), A269544 (r=7), A056799 (r=8), A056797 (r=9), A057462 (r=10), A251259 (r=11).

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

A383914 Primes p such that 12*2^p + 1 is also prime.

Original entry on oeis.org

3, 199, 3187, 44683, 59971, 213319, 303091, 916771

Offset: 1

Vincenzo Librandi, May 17 2025

If k is a term in A002253 and k-2 is prime, then k-2 is a term. - Amiram Eldar, May 17 2025

			3 is a term because 12*2^3+1 = 97 (prime).

Cf. A002253, A175172, A322301, A322302.

[p: p in PrimesUpTo (3500) | IsPrime(12*2^p+1)];

Select[Prime[Range[3500]],PrimeQ[12 2^#+1]&]

a(4)-a(8) from the b-file at A002253 added by Amiram Eldar, May 17 2025

cp's OEIS Frontend

A326655 Numbers k such that 3*4^k+1 is prime.

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

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Author

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PARI

Extensions

A383914 Primes p such that 12*2^p + 1 is also prime.

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Magma

Mathematica

Extensions

cp's OEIS Frontend

A326655 Numbers k such that 3*4^k+1 is prime.

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Author

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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 (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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A383914 Primes p such that 12*2^p + 1 is also prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

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.