A079706 -id:A079706 - cp's OEIS frontend

A084712 Smallest prime of the form (2n)^k + 1, or 0 if no such number exists.

3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37

Offset: 1

Amarnath Murthy, Jun 10 2003

It has not been proved that a(19), a(25), a(31), a(34), a(43) and a(46) are 0; if these values do exist, they have > 4000 digits. The other zeros are definite. - David Wasserman, Jan 03 2005
a((p-1)/2) = p for primes p > 2, or a(n) = 2n+1 for n = (p-1)/2. All other positive a(n) belong to A002496 = primes of form m^2 + 1. Corresponding positive exponents k are powers of 2. They are listed in A079706. - Alexander Adamchuk, Sep 17 2006
Because k must be a power of 2, numbers of the form (2n)^k+1 are called generalized Fermat numbers with base 2n. These numbers, like the regular Fermat numbers, are seldom prime. I checked n=19, 25, 31, 34, 43, 46 with k up to 2^16 without finding any primes. - T. D. Noe, May 13 2008
Comments from N. J. A. Sloane, Jan 27 2024: (Start)
As pointed out by Max Alekseyev, the previous version violated the OEIS rules, since a(19) has not been confirmed. I therefore removed the terms starting at a(19).
The previous DATA line read:
3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37, 0, 41, 43, 197352587024076973231046657, 47, 5308417, 0, 53, 2917, 3137, 59, 61, 0, 0, 67, 0, 71, 73, 5477, 1238846438084943599707227160577, 79, 40960001, 83, 7057, 0, 89
The old b-file has been changed to an a-file.
(End)

			a(7) = 197 = 14^2 + 1 as 14 + 1 = 15 is not a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..500 (currently known terms, may contain 0s in place of unknown terms)

Cf. A084713, A005097.

Table[k=1; While[p=1+(2n)^k; k<1024 && !PrimeQ[p], k=2k]; If[k==1024, 0, p], {n,44}] (* T. D. Noe, May 13 2008 *)

More terms from David Wasserman, Jan 03 2005

Edited by N. J. A. Sloane, Jan 27 2024 at the suggestion of Max Alekseyev

A255707 Least number k > 0 such that (2*n-1)^k - 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

0, 2, 1, 1, 1, 4, 1, 1, 6, 1, 1, 24, 1, 2, 2, 1, 1, 2, 2, 1, 4, 1, 1, 2, 1, 8, 4, 1, 12, 4, 1, 1, 8, 3, 1, 2, 1, 1, 2, 38, 1, 4, 1, 4, 2, 1, 2, 4, 747, 1, 4, 1, 1, 2, 1, 1, 10, 1, 2, 2, 2, 6, 42, 2, 1, 2, 1, 2, 10, 1, 1, 4, 2, 16, 50, 1, 1, 2, 22, 1, 2, 38

Offset: 1

Robert Price, Mar 02 2015

nonn

Michel Marcus, Table of n, a(n) for n = 1..152 (terms 1..143 from Robert Price)
Carlos Rivera, Puzzle 887. p(n)^c-2 is prime, The Prime Puzzles and Problems Connection.

Cf. A079706, A084712, A084713, A084714, A098090, A138066.

lst = {0}; For[n = 2, n ≤ 143, n++, For[k = 1, k >= 1, k++, If[PrimeQ[(2*n - 1)^k - 2], AppendTo[lst, k]; Break[]]]]; lst

a(n)=if(n==1,return(0));k=1;while(k,if(ispseudoprime((2*n-1)^k-2),return(k));k++)
vector(50,n,a(n)) \\ Derek Orr, Mar 03 2015

a(A098090(n)) = 1. - Michel Marcus, Mar 03 2015

A250200 Least number k>1 such that (2n-1)^k - 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

0, 2, 2, 2, 2, 4, 2, 2, 6, 2, 2, 24, 7, 2, 2, 3, 2, 2, 2, 4, 4, 2, 11, 2, 2, 8, 4, 2, 12, 4, 2, 2, 8, 3, 2, 2, 4, 2, 2, 38, 130, 4, 4, 4, 2, 3, 2, 4, 747, 3, 4, 2, 10, 2, 3, 17, 10, 13, 2, 2, 2, 6, 42, 2, 3, 2, 6, 2, 10, 2, 4, 4, 2, 16, 50, 3, 9, 2, 22, 25

Offset: 1

Robert Price, Mar 02 2015

nonn

Robert Price, Table of n, a(n) for n = 1..143

Cf. A084712, A084713, A084714, A079706, A138066, A255707.

lst = {0}; For[n = 2, n ≤ 143, n++, For[k = 2, k >= 1, k++, If[PrimeQ[(2*n - 1)^k - 2], AppendTo[lst, k]; Break[]]]]; lst
lnk[n_]:=Module[{k=2,c=2n-1},While[!PrimeQ[c^k-2],k++];k]; Join[{0}, Array[ lnk,80,2]] (* Harvey P. Dale, Jul 24 2017 *)

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

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

A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 1, -1, 0, 1, 0

Offset: 2

Eric Chen, Apr 19 2015

Least k such that the generalized Fermat number in base n (GFN(k,n)) is prime.
a(n) = -1 if n is in A070265 (perfect powers with an odd exponent).
a(n) is currently unknown for n = {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, ...}
Corresponding primes are {3, 2, 5, 3, 7, 1201, 0, 5, 11, 61, 13, 7, 197, 113, 17, 41761, 19, 181, 401, 11, 23, 139921, 577, 13, 677, 0, 29, 421, 31, ...}. (use 0 if a(n) = -1)
All 2 <= n <= 1500 and 0 <= k <= 14 are checked, the first occurrence of k (start with k = 0) in a(n) are {2, 11, 7, 43, 41, 75, 274, 234, 331, 1342, 824, ...}.

			a(7) = 2 since (7^(2^0)+1)/2 and (7^(2^1)+1)/2 are not primes, but (7^(2^2)+1)/2 = 1201 is prime.
a(14) = 1 since 14^(2^0)+1 is not prime, but 14^(2^1)+1 = 197 is prime.

Eric Chen, Table of n, a(n) for n = 2..1500 status (for the -1s, only a(n) for n in A070265 are proved, all other -1s are only conjectured)
Gary Barnes, List of generalized Fermat primes in even bases up to 1030
Eric Chen, List of generalized Fermat primes in bases up to 1000
Chris Caldwell, Generalized Fermat number
Richard Fischer, List of generalized Fermat primes in odd bases
Yves Gallot, Generalized Fermat prime search
Wilfrid Keller, Factorization of GFN(n,2)
Wilfrid Keller, Factorization of GFN(n,3)
Wilfrid Keller, Factorization of GFN(n,5)
Wilfrid Keller, Factorization of GFN(n,6)
Wilfrid Keller, Factorization of GFN(n,10)
Wilfrid Keller, Factorization of GFN(n,12)
Jeppe Stig Salling Nielsen, List of generalized Fermat primes in even bases up to 1000
MathWorld, Generalized Fermat number
OEIS wiki, Generalized Fermat number
Wikipedia, Generalized Fermat number

Cf. A079706, A228101, A084712, A123669, A058064, A057856, A130536, A080121, A077659, A122900.

Table[k=0; While[p=If[EvenQ[n], (2n)^(2^k)+1, ((2n)^(2^k)+1)/2]; k<12 && !PrimeQ[p], k=k+1]; If[k==12, -1, k], {n, 2, 1500}]

f(n) = for(k=0, 11, if(ispseudoprime(n^(2^k)+1), return(k))); -1
g(n) = for(k=0, 11, if(ispseudoprime((n^(2^k)+1)/2), return(k))); -1
a(n) = if(n%2==0, f(n), g(n))

f(n,k)=if(n%2, (n^(2^k)+1)/2, n^(2^k)+1)
a(n)=if(ispower(-n), -1, my(k); while(!ispseudoprime(f(n,k)), k++); k) \\ Charles R Greathouse IV, Apr 20 2015

a(2n) = A228101(n) = log_2(A079706(n)).

a(A006093(n)) = 0, a(A076274(n)) = 0, a(A070265(n)) = -1.

cp's OEIS Frontend

A084712 Smallest prime of the form (2n)^k + 1, or 0 if no such number exists.

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A255707 Least number k > 0 such that (2*n-1)^k - 2 is prime, or 0 if no such number exists.

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A250200 Least number k>1 such that (2n-1)^k - 2 is prime, or 0 if no such number exists.

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

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A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

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