A056993 -id:A056993 - cp's OEIS frontend

A253633 a(n) is the least positive integer b such that b^(2^n) + (b-1)^(2^n) is prime.

2, 2, 2, 2, 2, 9, 96, 32, 86, 60, 1079, 755, 312, 3509, 1829, 49958, 22845

Offset: 0

Jeppe Stig Nielsen, Jan 07 2015

When a(n) is 2, the corresponding prime is a Fermat prime, otherwise it is a so-called extended generalized Fermat prime sometimes denoted xGF(n, b, b-1) or similar.

			For n = 5, 2^5 = 32 is the exponent.  The numbers 1^32 + 0^32, 2^32 + 1^32, ..., 8^32 + 7^32 are not prime, but 9^32 + 8^32 is prime, so a(5) = 9. - _Michael B. Porter_, Mar 28 2018

Henri Lifchitz & Renaud Lifchitz, PRP Top Records, search for x^16384+y^16384, related to a(14).
Henri Lifchitz & Renaud Lifchitz, 49958^32768+49957^32768, a(15).
Henri Lifchitz & Renaud Lifchitz, 22845^65536+22844^65536, a(16).

Cf. A056993, A080208.

a(n)=for(b=2,10^10,if(ispseudoprime(b^(2^n)+(b-1)^(2^n)),return(b)))

a(n) = A080208(n) + 1.

a(13) from Jeppe Stig Nielsen, Mar 27 2018
a(14) found by Henri Lifchitz in 2007, from Jeppe Stig Nielsen, Apr 17 2018
a(15) found by Kellen Shenton, from Jeppe Stig Nielsen, Nov 27 2020
a(16) found by Kellen Shenton, from Jeppe Stig Nielsen, Mar 31 2021

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

A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

Original entry on oeis.org

1068630, 1441590, 1867950, 3429300, 4084230, 5651730, 6322890, 6770610, 7158630, 7804830, 9437760, 9624270, 13625850, 23194860, 25848840, 26588520, 28714950, 29451840, 32984430, 33650580, 36500910, 38177130, 42856590, 49531020, 50016540, 50222070, 52083330, 54637590

Offset: 1

Labos Elemer, May 07 2002

Numbers k such that C1(k) = k-1, C2(k) = k+1, C3(k) = k^2+k+1, C4(k) = k^2+1 and C5(k) = k^4+k^3+k^2+k+1 are all primes.

			For k = 1068630: the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials give a quintet of primes: {1068629, 1068631, 1141971145531, 1141970076901, 1304096876879617162402531}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A070155, A070156, A070157, A000068, A006313, A006314, A006315, A006316, A056993, A056994, A056995, A005574, A057465, A057002, A070020, A070025.

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1) && isprime(k^2+k+1) && isprime(k^4+k^3+k^2+k+1) ; \\ Amiram Eldar, Sep 24 2024

More terms from Don Reble, May 11 2002

a(24)-a(28) from Amiram Eldar, Sep 24 2024

A246121 Least k such that k^(6^n)*(k^(6^n) - 1) + 1 is prime.

Original entry on oeis.org

2, 3, 88, 28, 688, 7003, 1925

Offset: 0

Serge Batalov, Aug 14 2014

Numbers of the form k^m*(k^m - 1) + 1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors > 3. This sequence is a subset of A205506 with only m=6^n.

Numbers of this form are Generalized unique primes. a(6) generates a 306477-digit prime.

			When k = 88, k^72 - k^36 + 1 is prime. Since this isn't prime for k < 88, a(2) = 88.

C. Caldwell, Generalized unique primes

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246119, A246120.

a(n)=k=1; while(!ispseudoprime(k^(6^n)*(k^(6^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++)

a(n) = A085398(6^(n+1)). - Jinyuan Wang, Jan 01 2023

a(6) from Serge Batalov, Aug 15 2014

A228101 a(n) is the least k such that (2n)^(2^k) + 1 is a prime; a(n) = -1 if a prime (2n)^(2^k) + 1 is unknown, or = -2 if impossible.

Original entry on oeis.org

0, 0, 0, -2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, -2, 2, 0, -1, 0, 0, 4, 0, 2, -1, 0, 1, 1, 0, 0, -1, -2, 0, -1, 0, 0, 1, 4, 0, 2, 0, 1, -1, 0, 1, -1, 1, 0, -1, 0, 0, -1, 0, 0, 1, 0, 5, 1, 2, 1, -1, 1, 0, -2, 0, 2, 1, 0, 0, 2, 2, -1, 1, 0, 0, 3, 2, 0, 4, 1, 0, 2, 0

Offset: 1

Yves Gallot (galloty(AT)wanadoo.fr) and Robert G. Wilson v, Aug 14 2013

A prime number of the form b^(2^k) + 1 is called a generalized Fermat prime to base b.
See the hyperlink for more information and links.
The impossibility case, a(n) = -2, occurs exactly if 2n is a member of A070265. Or equivalently, n is in A126032. - Jeppe Stig Nielsen, Jul 02 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..500
Lucile and Yves Gallot, Welcome to the Generalized Fermat Prime Search!

Cf. A056993, A079706, A084712, A070265, A126032.

f[b_?EvenQ] := f[b] = Block[{k = 0}, While[! PrimeQ[b^(2^k) + 1], k++]; k];
lst = {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998}; (f[#] = -1) & /@ lst;
lst = {8, 32, 64, 128, 216, 512, 1000}; (f[#] = -2) & /@ lst; Table[ f[b], {b, 2, 1000, 2}]
(* Second program: *)
Module[{r = 83, nn = 12, s = {}, k}, Do[If[b > r, Break[], Do[If[Set[k, b^m/2] > r, Break[], AppendTo[s, k]], {m, 3, Infinity, 2}]], {b, 2, Infinity, 2}]; Table[If[MemberQ[s, n], -2, SelectFirst[Range[0, nn], PrimeQ[(2 n)^(2^#) + 1] &] /. x_ /; MissingQ@ x -> -1], {n, r}]] (* Michael De Vlieger, Jul 04 2017, Version 10.2 *)

Definition rewritten by Jeppe Stig Nielsen, Jul 02 2017

A275530 Smallest positive integer m such that (m^(2^n) + 1)/2 is prime.

Original entry on oeis.org

3, 3, 3, 9, 3, 3, 3, 113, 331, 513, 827, 799, 3291, 5041, 71, 220221, 23891, 11559, 187503, 35963

Offset: 0

Walter Kehowski, Jul 31 2016

The terms of this sequence with n > 11 correspond to probable primes which are too large to be proven prime currently. - Serge Batalov, Apr 01 2018

a(15) is a statistically significant outlier; the sequence (m^(2^15)+1)/2 may require a double-check with software that is not GWNUM-based. - Serge Batalov, Apr 01 2018

			a(7) = 113 since 113 is the smallest positive integer m such that (m^(2^7)+1)/2 is prime.

Richard Fischer, Generalized Fermat numbers with odd base
Wikipedia, Fermat number

Cf. A056993, A027862.

a:= proc(n) option remember; local m; for m by 2
      while not isprime((m^(2^n)+1)/2) do od; m
    end:
seq(a(n), n=0..8);

Table[m = 1; While[! PrimeQ[(m^(2^n) + 1)/2], m++]; m, {n, 0, 9}] (* Michael De Vlieger, Sep 23 2016 *)

a(n) = {my(m = 1); while (! isprime((m^(2^n)+1)/2), m += 2); m;} \\ Michel Marcus, Aug 01 2016

from sympy import isprime
def a(n):
  m, pow2 = 1, 2**n
  while True:
    if isprime((m**pow2 + 1)//2): return m
    m += 2
print([a(n) for n in range(9)]) # Michael S. Branicky, Mar 03 2021

a(13)-a(14) from Robert Price, Sep 23 2016
a(15) from Serge Batalov, Mar 29 2018
a(16) from Serge Batalov, Mar 30 2018
a(17) from Serge Batalov, Apr 01 2018
a(18)-a(19) from Ryan Propper, Aug 16 2022. These correspond to 1382288- and 2388581-digit PRPs, respectively, found using an exhaustive search with Jean Penne's LLR2.

A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 10, 6, 4, 2, 1, 12, 10, 6, 4, 2, 1, 16, 14, 16, 118, 44, 30, 1, 18, 16, 20, 132, 74, 54, 102, 1, 22, 20, 24, 140, 76, 96, 162, 120, 1, 28, 24, 28, 152, 94, 112, 274, 190, 278, 1, 30, 26, 34, 208, 156, 114, 300, 234, 614, 46, 1, 36, 36, 46, 240, 158

Offset: 0

Jeppe Stig Nielsen, Oct 01 2003

			{1}; {2,1}; {4,2,1}; ...
See the well-formed array on Gallot's page.

Harvey Dubner, J. Recr. Math., 18, 1986.

Yves Gallot, Generalized Fermat Prime Search.

Cf. A056993, A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362.

A122528 Minimal number k such that (2k)^(2^n) + 1 is prime, but (2k)^(2^m) + 1 is composite for m < n.

Original entry on oeis.org

1, 7, 17, 76, 22, 57, 137, 117, 307, 671, 412, 1279, 767, 35926, 50915, 35453, 24297, 114094, 12259, 37949, 459722

Offset: 0

Alexander Adamchuk, Sep 17 2006

A079706(a(n)) = 2^n which is the first occurrence of 2^n in A079706.

Corresponding primes A084712(a(n)) are {3, 197, 1336337, 284936905588473857, 197352587024076973231046657, ...}.

			a(0) = 1 because (2*1)^(2^0) + 1 = 2 + 1 = 3 is prime.
a(1) = 7 because (2*7)^(2^1) + 1 = 14^2 + 1 = 197 is prime but 14 + 1 = 15 is composite.

Yves Gallot et al., Generalized Fermat Prime Search
PrimeGrid, GFN Prime Search Status and History.

Cf. A079706, A084712.

Cf. A056993.

a(n)=for(k=1,+oo,if(ispseudoprime((2*k)^(2^n)+1),for(m=0,n-1,ispseudoprime((2*k)^(2^m)+1)&&next(2));return(k))) \\ Jeppe Stig Nielsen, Mar 10 2018

Definition corrected by T. D. Noe, May 14 2008
a(9) through a(16) from the extensive tables of generalized Fermat primes compiled by Yves Gallot and others. - T. D. Noe, May 14 2008
a(17)-a(20) from Jeppe Stig Nielsen, Mar 10 2018

A182154 Smallest k >= 2 such that k^(2^n)+1 is the lesser member of a twin prime pair.

Original entry on oeis.org

2, 2, 2, 4, 2, 49592, 7132, 532, 333482, 2226686, 3543554, 23379038, 1249625230, 188489906

Offset: 0

Manuel Valdivia, Apr 15 2012

These lesser of twin prime pairs are also generalized Fermat primes, (not possible for greater of twin prime pairs, except for 5).
When extending this sequence, it is useful if the primes b^(2^n)+1 are known in advance (Gallot link). - Jeppe Stig Nielsen, Sep 25 2019
For later terms, the bigger twin is only a probable prime, not a proven prime. - Jeppe Stig Nielsen, Nov 24 2022

			2^(2^4)+1 = 65537 = A001359(861), then a(4) = 2.

Yves Gallot, Generalized Fermat Prime Search.
OEIS Wiki, Generalized Fermat numbers.
PrimeGrid and "Stream", GFN-1x Small Primes search, mentions a(12) and a(13).

Cf. A056993, A001359.

Table[k=2; While[!PrimeQ[k^(2^n)+1]||!PrimeQ[k^(2^n)+3],k++]; k,{n,0,7}]

a(8)-a(10) from Jeppe Stig Nielsen, Sep 25 2019
Name edited by Felix Fröhlich, Sep 25 2019
a(11)-a(13) from Jeppe Stig Nielsen, Nov 24 2022

cp's OEIS Frontend

A253633 a(n) is the least positive integer b such that b^(2^n) + (b-1)^(2^n) is prime.

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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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A070042 At these values of k the 1st, 2nd, 3rd, 4th and 5th cyclotomic polynomials all give prime numbers.

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PARI

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A246121 Least k such that k^(6^n)*(k^(6^n) - 1) + 1 is prime.

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PARI

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A228101 a(n) is the least k such that (2n)^(2^k) + 1 is a prime; a(n) = -1 if a prime (2n)^(2^k) + 1 is unknown, or = -2 if impossible.

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A275530 Smallest positive integer m such that (m^(2^n) + 1)/2 is prime.

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A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

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