A070694 -id:A070694 - 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

A225560 Least numbers k for each base b >= 2 such that N = b^(2^n) + k is prime for 7 consecutive values from n = 0 to n = 6.

Original entry on oeis.org

66747, 18248, 53097, 2037018, 142531, 1691820, 1322535, 1659002, 266251, 6185640, 95075, 2518780, 657645, 325528, 71971, 2533260, 21494113, 682318, 3114879, 6523742, 9196027, 3588090, 12492473, 816078, 14837001, 12060370, 2933065, 12212058, 3122953

Offset: 2

Robin Garcia, May 10 2013

nonn

Generalized Fermat numbers of the form b^(2^n) + k

Cf. A070694, A225321, A225492

for(b=2,30,if(b%2==0,a=1,a=0);forstep(k=a,10^8,2,i=0;for(n=0,6,m=b^2^n+k;if(isprime(m),i++;if(i>6,print1(k,", ");next(3))))))

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

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.

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

A225321 Numbers b such that b^(2^n) + 3 is prime for n from 0 to 4.

Original entry on oeis.org

2564954, 4505138, 6319754, 10004666, 13410068, 28358686, 31079126, 31331314, 37983154, 40470296, 43452004, 58717498, 66643660, 67991588, 77422568, 77995658, 79257766, 98229376, 101553298, 123965218, 125464136, 126241688, 130818598, 130838170, 131305474

Offset: 1

Robin Garcia, May 05 2013

nonn

Generalized Fermat numbers of the form b^(2^n) + k with k = 3 here.

Cf. A070694

Select[Prime@Range[3, 10^6]-3, PrimeQ[#^2 + 3] && PrimeQ[#^4 + 3] && PrimeQ[#^8 + 3] && PrimeQ[#^16 + 3] &] (* Giovanni Resta, May 05 2013 *)

Missing terms a(2) and a(5) and a(12)-a(25) from Giovanni Resta, May 05 2013

A225392 Numbers b such that b^(2^n) + 2 is prime for n from 0 to 4.

Original entry on oeis.org

22155, 1864149, 2760681, 6222765, 22687797, 25631319, 29309589, 33333069, 36490905, 56310891, 60212889, 74097849, 76008207, 80864685, 84214395, 132006225, 132621171, 137362521, 138993381, 152223885, 154185525, 175950081, 188261481, 188677335, 194279955

Offset: 1

Robin Garcia, May 06 2013

nonn

Generalized Fermat numbers of the form b^(2^n) + k with k = 2 here.

For b = 2760681, b^(2^n) + 2 is prime for n from 0 to 5

Cf. A070694, A225321

k=2;forstep(b=1,5*10^7,[4,2,2,2],i=0;for(n=0,4,m=b^2^n+k;if(isprime(m),i++;if(i>4,print([b,m,n,i])))))

a(10)-a(25) from Giovanni Resta, May 06 2013

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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A225560 Least numbers k for each base b >= 2 such that N = b^(2^n) + k is prime for 7 consecutive values from n = 0 to n = 6.

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

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

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

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A225321 Numbers b such that b^(2^n) + 3 is prime for n from 0 to 4.

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A225392 Numbers b such that b^(2^n) + 2 is prime for n from 0 to 4.

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