A000215 -id:A000215 - cp's OEIS frontend

A059919 Generalized Fermat numbers: 3^(2^n)+1, n >= 0.

4, 10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962

Offset: 0

Henry Bottomley, Feb 08 2001

Generalized Fermat numbers (Ribenboim (1996))

F_n(a) := F_n(a,1) = a^(2^n) + 1, a >= 2, n >= 0, can't be prime if a is odd (as is the case for this sequence). - Daniel Forgues, Jun 19-20 2011

			a(0) = 3^(2^0)+1 = 3^1+1 = 4 = 2*(1)+2 = 2*(empty product)+2;
a(1) = 3^(2^1)+1 = 3^2+1 = 10 = 2*(4)+2;
a(2) = 3^(2^2)+1 = 3^4+1 = 82 = 2*(4*10)+2;
a(3) = 3^(2^3)+1 = 3^8+1 = 6562 = 2*(4*10*82)+2;
a(4) = 3^(2^4)+1 = 3^16+1 = 43046722 = 2*(4*10*82*6562)+2;
a(5) = 3^(2^5)+1 = 3^32+1 = 1853020188851842 = 2*(4*10*82*6562*43046722)+2;

Vincenzo Librandi, Table of n, a(n) for n = 0..10 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=3).
Wilfrid Keller, GFN3 factoring status.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n) + 1, n >= 0).
Cf. A059917 ((3^(2^n)+1)/2).
Cf. A199591, A078303, A078304, A152581, A080176, A199592, A152585.

[3^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

A059919:=n->3^(2^n)+1; seq(A059919(n), n=0..7); # Wesley Ivan Hurt, Jan 22 2014

Table[3^2^n + 1, {n, 0, 7}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

a(n) = { 3^(2^n) + 1 } \\ Harry J. Smith, Jun 30 2009

a(0) = 4; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = A011764(n)+1 = A059918(n+1)/A059918(n) = (A059917(n+1)-1)/(A059917(n)-1) = (A059723(n)/A059723(n+1))*(A059723(n+2)-A059723(n+1))/(A059723(n+1)-A059723(n))
a(n) = A057727(n)-1. - R. J. Mathar, Apr 23 2007
a(n) = 2*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 2*(empty product, i.e., 1) + 2 = 4 = a(0).
The above formula implies the GCD of any pair of terms is 2, which means that the terms of (3^(2^n)+1)/2 (A059917) are pairwise coprime. - Daniel Forgues, Jun 20 & 22 2011
Sum_{n>=0} 2^n/a(n) = 1/2. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 19 2011 and Jun 20 2011

A002586 Smallest prime factor of 2^n + 1.

Original entry on oeis.org

3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 65537, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 257, 3, 5, 3, 17, 3, 5, 3, 274177, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 65537, 3, 5, 3, 17, 3, 5

Offset: 1

N. J. A. Sloane

Conjecture: a(8+48*k) = 257 and a(40+48*k) = 257, where k is a nonnegative integer. - Thomas König, Feb 15 2017
Conjecture is true: 257 divides 2^(8+48*k)+1 and 2^(40+48*k)+1 but no prime < 257 ever does. Similarly, a(24+48*k) = 97. - Robert Israel, Feb 17 2017
From Robert Israel, Feb 17 2017: (Start)
If a(n) = p, there is some m such that a(n+m*j*n) = p for all j.
In particular, every member of the sequence occurs infinitely often.
a(k*n) <= a(n) for any odd k. (End)

			a(2^k) = 3, 5, 17, 257, 65537 is the k-th Fermat prime 2^(2^k) + 1 = A019434(k) for k = 0, 1, 2, 3, 4. - _Jonathan Sondow_, Nov 28 2012

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres, Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of n, a(n) for n = 1..3967 (first 500 terms from T. D. Noe)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Thomas König, C program using gmp for testing the conjectures that a(8+k*48) = 257 and a(40+k*48) = 257
E. Lucas, Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A000215, A001269, A002587, A019434, A050922, A093179.

f[n_] := FactorInteger[2^n + 1][[1, 1]]; Array[f, 100] (* Robert G. Wilson v, Nov 28 2012 *)
FactorInteger[#][[1,1]]&/@(2^Range[90]+1) (* Harvey P. Dale, Jul 25 2024 *)

a(n) = my(m=n%8); if(m, [3, 5, 3, 17, 3, 5, 3][m], factor(2^n+1)[1,1]); \\ Ruud H.G. van Tol, Feb 16 2024

from sympy import primefactors
smallest_primef = []
for n in range(1,87):
    y = (2 ** n) + 1
    smallest_primef.append(min(primefactors(y)))
print(smallest_primef) # Adrienne Leonardo, Dec 29 2024

a(n) = 3, 5, 3, 17, 3, 5, 3 for n == 1, 2, 3, 4, 5, 6, 7 (mod 8). (Proof. Let n = k*odd with k = 1, 2, or 4. As 2^k = 2, 4, 16 == -1 (mod 3, 5, 17), we get 2^n + 1 = 2^(k*odd) + 1 = (2^k)^odd + 1 == (-1)^odd + 1 == 0 (mod 3, 5, 17). Finally, 2^n + 1 !== 0 (mod p) for prime p < 3, 5, 17, respectively.) - Jonathan Sondow, Nov 28 2012

More terms from James Sellers, Jul 06 2000

Definition corrected by Jonathan Sondow, Nov 27 2012

A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

Original entry on oeis.org

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 0

N. J. A. Sloane

The 32 divisors of the product of the 5 known Fermat primes.
The only known odd numbers whose totient is a power of 2. - Labos Elemer, Dec 06 2000
Equals first 32 members of A001317. Also, equals first 32 members of A053576. - Omar E. Pol, Dec 10 2008
Omitting the first term a(0)=1 gives A045544 (the number of sides of constructible odd-sided regular polygons.)

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 140.

Eric Weisstein's World of Mathematics, Regular Polygon, Sierpiński Sieve, Constructible Polygon
OEIS Wiki, Constructible odd-sided polygons
OEIS Wiki, Sierpinski's triangle
Index entries for sequences related to divisors of numbers

Essentially same as A045544.

Cf. A000010, A000215, A001317, A003401, A003527, A004169, A004729, A019434, A045544, A047999, A053576, A054432.

Mathematica
```
Divisors[2^32-1]
```
PARI
```
divisors(1<<32-1)
```

Edited by Daniel Forgues, Jun 17 2011

A080176 Generalized Fermat numbers: 10^(2^n) + 1, n >= 0.

Original entry on oeis.org

11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001

Offset: 0

Jens Voß, Feb 04 2003

As for standard Fermat numbers 2^(2^n) + 1, a number (2b)^m + 1 (with b > 1) can only be prime if m is a power of 2. On the other hand, out of the first 12 base-10 Fermat numbers, only the first two are primes.

Also, binary representation of Fermat numbers (in decimal, see A000215).

			a(0) = 10^1 + 1 = 11 = 9*(1) + 2 = 9*(empty product) + 2.
a(1) = 10^2 + 1 = 101 = 9*(11) + 2.
a(2) = 10^4 + 1 = 10001 = 9*(11*101) + 2.
a(3) = 10^8 + 1 = 100000001 = 9*(11*101*10001) + 2.
a(4) = 10^16 + 1 = 10000000000000001 = 9*(11*101*10001*100000001) + 2.
a(5) = 10^32 + 1 = 100000000000000000000000000000001 = 9*(11*101*10001*100000001*10000000000000001) + 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..9 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=10).
Wilfrid Keller, GFN10 factoring status.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n) + 1, n >= 0).

Cf. A019434, A080174, A080175, A059919, A199591, A078303, A078304, A152581, A199592, A152585.

[10^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Table[10^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

a(0) = 11; a(n) = (a(n - 1) - 1)^2 + 1.
a(n) = 9*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 9*(empty product, i.e., 1)+ 2 = 11 = a(0). - Daniel Forgues, Jun 20 2011
Sum_{n>=0} 2^n/a(n) = 1/9. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 19 2011

A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

7, 25, 39, 75, 127, 1947, 3313, 23473, 125413

Offset: 1

Arkadiusz Wesolowski, Jun 05 2013

No other terms below 5330000.

The reason all terms are odd is that if k is even, then 5*2^k + 1 == (-1)*(-1)^k + 1 = (-1)*1 + 1 = 0 (mod 3). So if k is even, then 3 divides 5*2^k + 1, and since 3 divides no other Fermat number than F_0=3 itself, we do not have a Fermat factor. - Jeppe Stig Nielsen, Jul 21 2019

Wilfrid Keller, Fermat factoring status
J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002254.

Cf. A000215, A050526, A057775, A057778, A201364, A204620.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 3313, 2}]; lst

isok(n) = my(p = 5*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ Michel Marcus, Nov 10 2018

A081092 Primes having a prime number of 1's in their binary representation.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443

Offset: 1

Reinhard Zumkeller, Mar 05 2003

Same as primes with prime binary digit sum.
Primes with prime decimal digit sum are A046704.
Sum_{a(n) < x} 1/a(n) is asymptotic to log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 09 2012
A049084(A000120(a(n))) > 0; A081091, A000215 and A081093 are subsequences.

			15th prime = 47 = '101111' with five 1's, therefore 47 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Cf. A000040, A000120, A046704, A081093.

Subsequence of A052294.

a081092 n = a081092_list !! (n-1)
a081092_list = filter ((== 1) . a010051') a052294_list
-- Reinhard Zumkeller, Nov 16 2012

q:= n-> isprime(n) and isprime(add(i,i=Bits[Split](n))):
select(q, [$1..500])[];  # Alois P. Heinz, Sep 28 2023

Clear[BinSumOddQ];BinSumPrimeQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];PrimeQ[s]]; lst={};Do[p=Prime[n];If[BinSumPrimeQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime[Range[100]], PrimeQ[Apply[Plus, IntegerDigits[#, 2]]] &] (* Jonathan Sondow, Jun 09 2012 *)

lista(nn) = {forprime(p=2, nn, if (isprime(hammingweight(p)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2015

from sympy import isprime
def ok(n): return isprime(n.bit_count()) and isprime(n)
print([k for k in range(444) if ok(k)]) # Michael S. Branicky, Dec 27 2023

A093179 Smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.

Original entry on oeis.org

3, 5, 17, 257, 65537, 641, 274177, 59649589127497217, 1238926361552897, 2424833, 45592577, 319489, 114689, 2710954639361, 116928085873074369829035993834596371340386703423373313, 1214251009, 825753601, 31065037602817, 13631489, 70525124609

Offset: 0

Eric W. Weisstein, Mar 27 2004

a(14) might need to be corrected if F(14) turns out to have a smaller factor than 116928085873074369829035993834596371340386703423373313. F(20) is composite, but no explicit factor is known. - Jeppe Stig Nielsen, Feb 11 2010

			F(0) = 2^(2^0) + 1 = 3, prime.
F(5) = 2^(2^5) + 1 = 4294967297 = 641*6700417.
So 3 as the 0th entry and 641 is the 5th term.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 73.

Wilfrid Keller, Prime factors k·2^n + 1 of Fermat numbers F_m and complete factoring status
Ivars Peterson, Cracking Fermat Numbers
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000051, A000215, A020639, A070592, A007117.

Leading entries in triangle A050922.

Table[With[{k = 2^n}, FactorInteger[2^k + 1]][[1, 1]], {n, 0, 15, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

g(n)=for(x=9,n,y=Vec(ifactor(2^(2^x)+1));print1(y[1]",")) \\ Cino Hilliard, Jul 04 2007

a(n) = A007117(n)*2^(n+2) + 1 for n >= 2. - Jianing Song, Mar 02 2021

a(n) = A020639(A000215(n)). - Alois P. Heinz, Oct 25 2024

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar
a(14)-a(15) added by Jeppe Stig Nielsen, Feb 11 2010
a(16)-a(19) added based on terms of A007117 by Jianing Song, Mar 02 2021

A245970 Tower of 2's modulo n.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 2, 0, 7, 6, 9, 4, 3, 2, 1, 0, 1, 16, 5, 16, 16, 20, 6, 16, 11, 16, 7, 16, 25, 16, 2, 0, 31, 18, 16, 16, 9, 24, 16, 16, 18, 16, 4, 20, 16, 6, 17, 16, 23, 36, 1, 16, 28, 34, 31, 16, 43, 54, 48, 16, 22, 2, 16, 0, 16, 64, 17, 52, 52, 16, 3, 16

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (2^(2^(2^(2^(2^ ... ))))) mod n, provided enough 2's are in the tower so that adding more doesn't affect the value of a(n).

Let b(i) = A014221(i) = (2^(2^(2^(2^(2^ ... ))))), with i 2's. Since gcd(b(i)+1, b(j)+1) = gcd(2^2^b(i-2)+1, 2^2^b(j-2)+1) = gcd(A000215(b(i-2)), A000215(b(j-2))) = 1 for 1 <= i < j, there is no n > 1 such that a(n) = n-1. Since b(i)-1 = 2^2^b(i-2)-1 divides b(j)-1 = 2^2^b(j-2)-1 for 1 <= i < j, a(n) = 1 if and only if n > 1 is a divisor of a number of the form b(i)-1, or if and only if n > 1 is a divisor of a Fermat number (A023394). - Jianing Song, May 16 2024

			a(5) = 1, as 2^x mod 5 is 1 for x being any even multiple of two and X = 2^(2^(2^...)) is an even multiple of two.

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A014221, A027763, A240162, A245971, A245972, A245973, A245974, A000010, A000079.

import Math.NumberTheory.Moduli (powerMod)
a245970 n = powerMod 2 (phi + a245970 phi) n
            where phi = a000010 n
-- Reinhard Zumkeller, Feb 01 2015

A:= proc(n)
     local phin,F,L,U;
     phin:= numtheory:-phi(n);
     if phin = 2^ilog2(phin) then
        F:= ifactors(n)[2];
        L:= map(t -> t[1]^t[2],F);
        U:= [seq(`if`(F[i][1]=2,0,1),i=1..nops(F))];
        chrem(U,L);
     else
        2 &^ A(phin) mod n
     fi
end proc:
seq(A(n), n=2 .. 100); # Robert Israel, Aug 19 2014

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[2, 2^n, n] (* 2^^(2^n) (mod n), in Knuth's up-arrow notation *); Array[f, 72]
(* Second program: *)
a[n_] := Module[{phin, F, L, U},
   phin = EulerPhi[n];
   If[phin == 2^Floor@Log2[phin],
      F = FactorInteger[n];
      L = Power @@@ F;
      U = Table[If[F[[i, 1]] == 2, 0, 1], {i, 1, Length[F]}];
      ChineseRemainder[U, L],
      (2^a[phin])~Mod~n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 03 2023, after Robert Israel *)

a(n)=if(n<3, return(0)); my(e=valuation(n,2),k=n>>e); lift(chinese(Mod(2,k)^a(eulerphi(k)), Mod(0,2^e))) \\ Charles R Greathouse IV, Jul 29 2016

def tower2mod(n):
    if ( n <= 22 ):
        return 65536%n
    else:
        ep = euler_phi(n)
        return power_mod(2,ep+tower2mod(ep),n)

a(n) = 2^(A000010(n)+a(A000010(n))) mod n.
a(n) = 0 if n is a power of 2.
a(n) = (2^2) mod n, if n < 5.
a(n) = (2^(2^2)) mod n, if n < 11.
a(n) = (2^(2^(2^2))) mod n, if n < 23.
a(n) = (2^(2^(2^(2^2)))) mod n, if n < 47.
a(n) = (2^^k) mod n, if n < A027763(k), where ^^ is Knuth's double-arrow notation.
From Robert Israel, Aug 19 2014: (Start)
If gcd(m,n) = 1, then a(m*n) is the unique k in [0,...,m*n-1] with
k == a(n) mod n and k == a(m) mod m.
a(n) = 1 if n is a Fermat number.
a(n) = 2^a(A000010(n)) mod n if n is not in A003401.
(End)

A078303 Generalized Fermat numbers: 6^(2^n) + 1, n >= 0.

Original entry on oeis.org

7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097

Offset: 0

Eric W. Weisstein, Nov 21 2002

The next term is too large to include.
As for standard Fermat numbers 2^(2^n) + 1, a number (2b)^m + 1 (with b > 1) can only be prime if m is a power of 2. On the other hand, out of the first 13 base-6 Fermat numbers, only the first three are primes.
Either the sequence of (standard) Fermat numbers contains infinitely many composite numbers or the sequence of base-6 Fermat numbers contains infinitely many composite numbers (cf. https://mathoverflow.net/a/404235/1593). - José Hernández, Nov 09 2021
Since all powers of 6 are congruent to 6 (mod 10), all terms of this sequence are congruent to 7 (mod 10). - Daniel Forgues, Jun 22 2011
There are only 5 known Fermat primes of the form 2^(2^n) + 1: {3, 5, 17, 257, 65537}. There are only 2 known base-10 generalized Fermat primes of the form 10^(2^n) + 1: {11, 101}. - Alexander Adamchuk, Mar 17 2007

			a(0) = 6^1+1 = 7 = 5*(1)+2 = 5*(empty product)+2;
a(1) = 6^2+1 = 37 = 5*(7)+2;
a(2) = 6^4+1 = 1297 = 5*(7*37)+2;
a(3) = 6^8+1 = 1679617 = 5*(7*37*1297)+2;
a(4) = 6^16+1 = 2821109907457 = 5*(7*37*1297*1679617)+2;
a(5) = 6^32+1 = 7958661109946400884391937 = 5*(7*37*1297*1679617*2821109907457)+2;

Vincenzo Librandi, Table of n, a(n) for n = 0..12
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=6).
Wilfrid Keller, GFN06 factoring status.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n) + 1, n >= 0).
Cf. A019434 (Fermat primes of the form 2^(2^n) + 1).
Cf. A123669, A123599, A056993, A126032, A178428, A059919, A199591, A078304, A152581, A080176, A199592, A152585.

[6^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Table[6^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

a(n)=6^(2^n)+1 \\ Charles R Greathouse IV, Jun 21 2011

a(0) = 7, a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = 5*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 5*(empty product, i.e., 1)+ 2 = 7 = a(0). This implies that the terms are pairwise coprime. - Daniel Forgues, Jun 20 2011
Sum_{n>=0} 2^n/a(n) = 1/5. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 22 2011

A078304 Generalized Fermat numbers: 7^(2^n)+1, n >= 0.

Original entry on oeis.org

8, 50, 2402, 5764802, 33232930569602, 1104427674243920646305299202, 1219760487635835700138573862562971820755615294131238402

Offset: 0

Eric W. Weisstein, Nov 21 2002

From Daniel Forgues, Jun 19 2011: (Start)
Generalized Fermat numbers F_n(a) := F_n(a,1) = a^(2^n)+1, a >= 2, n >= 0, can't be prime if a is odd (as is the case for the current sequence) (Ribenboim (1996)).
All factors of generalized Fermat numbers F_n(a,b) := a^(2^n)+b^(2^n), a >= 2, n >= 0, are of the form k*2^m+1, k >= 1, m >=0 (Riesel (1994, 1998)). (This only expresses that the factors are odd, which means that it only applies to odd generalized Fermat numbers.) (End)

			a(0) = 7^1+1 = 8 = 6*(1)+2 = 6*(empty product)+2.
a(1) = 7^2+1 = 50 = 6*(8)+2.
a(2) = 7^4+1 = 2402 = 6*(8*50)+2.
a(3) = 7^8+1 = 5764802 = 6*(8*50*2402)+2.
a(4) = 7^16+1 = 33232930569602 = 6*(8*50*2402*5764802)+2.
a(5) = 7^32+1 = 1104427674243920646305299202 = 6*(8*50*2402*5764802*33232930569602)+2.

Vincenzo Librandi, Table of n, a(n) for n = 0..12
Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n)+1, n >= 0).

Cf. A059919, A199591, A078303, A152581, A080176, A199592, A152585.

[7^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Table[7^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

a(0) = 8, a(n)=(a(n-1)-1)^2+1, n >= 1.
a(n) = 6*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 6*(empty product, i.e., 1)+ 2 = 8 = a(0). This means that the GCD of any pair of terms is 2. - Daniel Forgues, Jun 20 2011
Sum_{n>=0} 2^n/a(n) = 1/6. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 19 2011

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