A335805 -id:A335805 - 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

A235390 Numbers k such that k^(2^i)+1 are primes for i=0...5.

Original entry on oeis.org

1, 7072833120, 9736020616, 12852419340, 36632235070, 41452651506, 44619665520, 53569833730, 54673378956, 66032908020, 69449109580, 69936419290, 82549220670, 99574135650, 106362659256, 108208833756, 113366066976, 136032409906, 167385272500, 174963279540, 195763339776

Offset: 1

Alex Ratushnyak, Jan 09 2014

nonn

A subsequence of A070694.
Conjecture: the sequence is infinite.
For n=4 and n=9, a(n)*2+1 is also a prime.
The first term greater than 1 such that k^(2^6) + 1 is also prime, is a(148) = 2072005925466, see A335805. - Jeppe Stig Nielsen, Aug 18 2020

			k=7072833120 is in the sequence because the following are six primes: 7072833121, 7072833120^2+1, k^4+1, k^8+1, k^16+1, k^32+1.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..101 (calculated by Yves Gallot (pers. communication), terms n = 2..94 from Martin Raab)
Yves Gallot, GFP (Generalized Fermat Progressions) / gfp6, software for calculating this sequence.

Cf. A000040, A006093, A019434, A056993, A070325, A070655, A070689, A070694, A090872, A335805.

a(1)=1 inserted by Jeppe Stig Nielsen, Aug 11 2020

A337364 Numbers b such that b^(2^i) + 1 is prime for i = 0...7.

Original entry on oeis.org

1, 240164550712338756, 3686834112771042790, 6470860179642426900, 7529068955648085700, 10300630358100537120, 16776829808789151280, 17622040391833711780, 19344979062504927000, 23949099004395080026, 25348938242408650240, 30262840543567048476, 35628481193915651646

Offset: 1

Jeppe Stig Nielsen, Aug 25 2020

nonn

Explicitly, for each b, the eight numbers b+1, b^2+1, b^4+1, b^8+1, b^16+1, b^32+1, b^64+1, and b^128+1 must be primes (generalized Fermat primes).

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..31 (found by Rob Gahan)
Yves Gallot, GFP (Generalized Fermat Progressions) / gfp8, software for calculating this sequence.

Cf. A006093, A019434, A056993, A070325, A070655, A070689, A070694, A090872, A235390, A335805.

a(10)-a(12) from Jeppe Stig Nielsen, Sep 04 2020

a(13) found by Rob Gahan added by Jeppe Stig Nielsen, Feb 15 2021

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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A235390 Numbers k such that k^(2^i)+1 are primes for i=0...5.

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A337364 Numbers b such that b^(2^i) + 1 is prime for i = 0...7.

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