A337364 -id:A337364 - cp's OEIS frontend

A090872 a(n) is the smallest number m greater than 1 such that m^(2^k)+1 for k=0,1,...,n are primes.

2, 2, 2, 2, 2, 7072833120, 2072005925466, 240164550712338756

Offset: 0

Farideh Firoozbakht, Jan 31 2004

The first five terms of this sequence correspond to Fermat primes.

Note that 7072833120 is not the smallest base to give at least six possibly nonconsecutive k values. For example, 292582836^(2^k) + 1 is prime for k = 0,1,2,3,4,7. - Jeppe Stig Nielsen, Sep 18 2022

			a(5)=7072833120 because 7072833120^2^k+1 for k=0,1,2,3,4,5 are primes.

PrimeGrid, Generalized Fermat Progression Search
Carlos Rivera, Puzzle 137. Product of primes + 1, a square, The Prime Puzzles and Problems Connection.
Carlos Rivera, Prime Puzzle 399. Some more terms, The Prime Puzzles and Problems Connection.

Cf. A019434, A000215.
Cf. A090873, A090874, A090875.
All solutions for fixed n: A006093 (n=0), A070689 (n=1), A070325 (n=2), A070655 (n=3), A070694 (n=4), A235390 (n=5), A335805 (n=6), A337364 (n=7).

a(6) from Jens Kruse Andersen, May 06 2007

a(7) from Kellen Shenton, Aug 13 2020

A335805 Numbers b such that b^(2^i) + 1 is prime for i = 0...6.

Original entry on oeis.org

1, 2072005925466, 5082584069416, 12698082064890, 29990491969260, 46636691707050, 65081025897426, 83689703895606, 83953213480290, 105003537341346, 105699143244090, 107581715369910, 111370557491826, 111587899569066, 128282713771996, 133103004825210

Offset: 1

Jeppe Stig Nielsen, Aug 14 2020

nonn

Explicitly, for each b, the seven numbers b+1, b^2+1, b^4+1, b^8+1, b^16+1, b^32+1, and b^64+1 must be primes (generalized Fermat primes).

The first term greater than 1 such that b^(2^7) + 1 is also prime, is 240164550712338756, see A337364. - Jeppe Stig Nielsen, Aug 25 2020

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..4603 (up to a(n) = A337364(2), calculated by _Kellen Shenton_)
Yves Gallot, GFP (Generalized Fermat Progressions) / gfp7, software for calculating this sequence.

Cf. A006093, A019434, A056993, A070325, A070655, A070689, A070694, A090872, A235390.

A336364 Rectangular array by antidiagonals: row n shows the positive integers whose distance to the nearest prime is n.

Original entry on oeis.org

2, 3, 1, 5, 4, 9, 7, 6, 15, 26, 11, 8, 21, 34, 93, 13, 10, 25, 50, 117, 118, 17, 12, 27, 56, 123, 122, 119, 19, 14, 33, 64, 143, 144, 121, 120, 23, 16, 35, 76, 145, 186, 205, 300, 531, 29, 18, 39, 86, 185, 204, 217, 324, 533, 532, 31, 20, 45, 92, 187, 206

Offset: 1

Clark Kimberling, Jul 19 2020

Row 1: the primes, A000040. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

			Corner:
   2   3   5   7  11   13   17   19   23   29   31   37
   1   4   6   8  10   12   14   16   18   20   22   24
   9  15  21  25  27   33   35   39   45   49   51   55
  26  34  50  56  64   76   86   92   94  116  124  134
  93 117 123 143 145  185  187  203  207  215  219  245

Sean A. Irvine, Java program (github).
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A051699, A336365.

a[?PrimeQ] = 0; a[n] := Min[NextPrime[n] - n, n - NextPrime[n, -1]];
t = Table[a[n], {n, 1, 2000}]; (* A051699 *)
r[n_] := Flatten[Position[t, n]]; u[n_, k_] := r[n][[k]];
TableForm[Table[u[n, k], {n, 0, 15}, {k, 1, Length[r[n]]}]] (* A337364, array *)
Table[u[n - k, k], {n, 0, 15}, {k, n, 1, -1}] // Flatten    (* A337364, sequence *)

cp's OEIS Frontend

A090872 a(n) is the smallest number m greater than 1 such that m^(2^k)+1 for k=0,1,...,n are primes.

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A335805 Numbers b such that b^(2^i) + 1 is prime for i = 0...6.

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A336364 Rectangular array by antidiagonals: row n shows the positive integers whose distance to the nearest prime is n.

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Mathematica