A070325 -id:A070325 - 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

A070655 Numbers k such that k+1, k^2+1, k^4+1 and k^8+1 are primes.

Original entry on oeis.org

1, 2, 4, 19380, 285090, 337536, 448630, 532390, 534430, 545140, 547536, 585106, 602056, 677076, 876180, 1007386, 1030200, 1331950, 1462000, 1736346, 1878790, 1883856, 2071960, 2194666, 2240890, 2763010, 2824720, 3018606, 3114996

Offset: 1

Robert G. Wilson v, May 13 2002

nonn

Jason Bard, Table of n, a(n) for n = 1..10000

Cf. A070325.

Do[ If[ PrimeQ[n + 1] && PrimeQ[n^2 + 1] && PrimeQ[n^4 + 1] && PrimeQ[n^8 + 1], Print[n]], {n, 1, 10^7}]

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

A237191 Numbers k such that k+1, k+3, k^2+1, k^2+3, k^4+1, k^4+3 are six primes.

Original entry on oeis.org

2, 520360, 14320216, 30527896, 119668186, 120506050, 131448430, 142493926, 211158676, 254574706, 276368680, 306216940, 315122416, 421132180, 472731400, 506213890, 540271396, 616078786, 629310346, 646308250, 741176296, 752897860, 800587480, 851425030, 897745996

Offset: 1

Alex Ratushnyak, Feb 04 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000

A subsequence of A067662, A070325, A070689, A080149.

from sympy import isprime
for n in range(0,1000000000,2):
    if isprime(n+1) and isprime(n*n+1) and isprime(n**4+1):
        if isprime(n+3) and isprime(n*n+3) and isprime(n**4+3):
            print(n, end=', ')

A188698 Numbers k such that 1+k, 1+k^2, 1+k^4 and 1+k^16 are all prime.

Original entry on oeis.org

1, 2, 690, 33190, 57106, 77140, 135606, 258990, 303430, 331140, 337536, 359230, 375646, 455526, 458326, 493396, 548226, 550540, 585106, 602056, 659250, 680830, 742306, 800406, 827680, 870240, 918340, 925390, 968320, 1203100, 1273890, 1455526, 1497576, 1605016

Offset: 1

Zak Seidov, Apr 09 2011

nonn

Subsequence of A070325, which itself is a subsequence of A070689, which itself is a subsequence of A006093.

			a(3) = 690 = A070689(32) = A070325(11) = A006093(125).

[n: n in [0..7000000]| IsPrime(1+n) and IsPrime(1+n^2) and IsPrime(1+n^4) and IsPrime(1+n^16)]; // Vincenzo Librandi, Apr 11 2011

a(1) = 1 prepended by Vincenzo Librandi, Apr 11 2011

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A070655 Numbers k such that k+1, k^2+1, k^4+1 and k^8+1 are primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A237191 Numbers k such that k+1, k+3, k^2+1, k^2+3, k^4+1, k^4+3 are six primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A188698 Numbers k such that 1+k, 1+k^2, 1+k^4 and 1+k^16 are all prime.

Views

Author

Keywords

Comments

Examples

Programs

Magma

Extensions