A152585 -id:A152585 - cp's OEIS frontend

A178426 11 followed by the Fermat numbers A152585.

11, 13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137

Offset: 0

Roger L. Bagula, May 27 2010

If a(0)=3, the recursion formula gives A000215.

Cf. A000215, A178427, A178428.

a[0] := 11;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] + 2;
Table[a[n], {n, 0, 10}]

a(0)=11. a(n)= 2 + Product_{i=0..n-1} a(i).

Definition simplified by the Assoc. Eds. of the OEIS - May 28 2010

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

Original entry on oeis.org

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

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

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

A199591 Generalized Fermat numbers: 5^(2^n) + 1, n >= 0.

Original entry on oeis.org

6, 26, 626, 390626, 152587890626, 23283064365386962890626, 542101086242752217003726400434970855712890626

Offset: 0

Arkadiusz Wesolowski, Nov 08 2011

			a(0) = 5^(2^0) + 1 = 5^1 + 1 = 6 = 4*(2^0) + 2;
a(1) = 5^(2^1) + 1 = 5^2 + 1 = 26 = 4*(2^1*3) + 2;
a(2) = 5^(2^2) + 1 = 5^4 + 1 = 626 = 4*(2^2*3*13) + 2;
a(3) = 5^(2^3) + 1 = 5^8 + 1 = 390626 = 4*(2^3*3*13*313) + 2;
a(4) = 5^(2^4) + 1 = 5^16 + 1 = 152587890626 = 4*(2^4*3*13*313*195313) + 2;
a(5) = 5^(2^5) + 1 = 5^32 + 1 = 23283064365386962890626 = 4*(2^5*3*13*313*195313*76293945313) + 2;

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..11
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=5).
Wilfrid Keller, GFN05 factoring status.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A059919, A078303, A078304, A152581, A080176, A199592, A152585.

Magma
```
[5^2^n+1 : n in [0..6]];
```
Mathematica
```
Table[5^2^n + 1, {n, 0, 6}]
```
PARI
```
for(n=0, 6, print1(5^2^n+1, ", "))
```

a(0) = 6; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(0) = 6, a(1) = 26; a(n) = a(n-1) + 4*5^(2^(n-1))*Product_{i=0..n-2} a(i), n >= 2.
a(0) = 6, a(1) = 26; a(n) = a(n-1)^2 - 2*(a(n-2)-1)^2, n >= 2.
a(0) = 6; a(n) = 4*(Product_{i=0..n-1} a(i)) + 2, n >= 1.
a(n) = A152578(n) - 1.
Sum_{n>=0} 2^n/a(n) = 1/4. - Amiram Eldar, Oct 03 2022

A199592 Generalized Fermat numbers: 11^(2^n) + 1, n >= 0.

Original entry on oeis.org

12, 122, 14642, 214358882, 45949729863572162, 2111377674535255285545615254209922, 4457915684525902395869512133369841539490161434991526715513934826242

Offset: 0

Arkadiusz Wesolowski, Nov 08 2011

			a(0) = 11^(2^0) + 1 = 11^1 + 1 = 12 = 10*(2^0) + 2;
a(1) = 11^(2^1) + 1 = 11^2 + 1 = 122 = 10*(2^1*6) + 2;
a(2) = 11^(2^2) + 1 = 11^4 + 1 = 14642 = 10*(2^2*6*61) + 2;
a(3) = 11^(2^3) + 1 = 11^8 + 1 = 214358882 = 10*(2^3*6*61*7321) + 2;
a(4) = 11^(2^4) + 1 = 11^16 + 1 = 45949729863572162 = 10*(2^4*6*61*7321*107179441) + 2;
a(5) = 11^(2^5) + 1 = 11^32 + 1 = 2111377674535255285545615254209922 = 10*(2^5*6*61*7321*107179441*22974864931786081) + 2;

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..11
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. A059919, A199591, A078303, A078304, A152581, A080176, A152585.

Magma
```
[11^2^n+1 : n in [0..6]]
```
Mathematica
```
Table[11^2^n + 1, {n, 0, 6}]
```
PARI
```
for(n=0, 6, print1(11^2^n+1, ", "))
```

a(0) = 12; a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(0) = 12, a(1) = 122; a(n) = a(n-1) + 10*11^(2^(n-1))*Product_{i=0..n-2} a(i), n >= 2.
a(0) = 12, a(1) = 122; a(n) = a(n-1)^2 - 2*(a(n-2)-1)^2, n >= 2.
a(0) = 12; a(n) = 10*(Product_{i=0..n-1} a(i)) + 2, n >= 1.
a(n) = A152583(n) - 1.
Sum_{n>=0} 2^n/a(n) = 1/10. - Amiram Eldar, Oct 03 2022

A273950 Prime factors of generalized Fermat numbers of the form 12^(2^m) + 1 with m >= 0.

Original entry on oeis.org

5, 13, 17, 29, 89, 97, 233, 257, 769, 36097, 40961, 65537, 81281, 153953, 163841, 260753, 1724417, 4550657, 5767169, 8253953, 11304961, 13631489, 21495809, 69619841, 77651969, 147849217, 158334977, 159522817, 1711276033, 6528575489, 27286044673, 52613349377

Offset: 1

Arkadiusz Wesolowski, Jun 05 2016

nonn

Primes p such that the multiplicative order of 12 (mod p) is a power of 2.

Hans Riesel, Common prime factors of the numbers A_n=a^(2^n)+1, BIT 9 (1969), pp. 264-269.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..44
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, pp. 397-405.
OEIS Wiki, Generalized Fermat numbers

Cf. A023394, A072982, A152585, A268660, A268664, A273945 (base 3), A273946 (base 5), A273947 (base 6), A273948 (base 7), A273949 (base 11).

Select[Prime@Range[2, 10^5], IntegerQ@Log[2, MultiplicativeOrder[12, #]] &]

A268660 Numbers n such that 3*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Original entry on oeis.org

2, 5, 8, 41, 209, 353, 2816, 20909, 42665, 157169, 213321, 303093, 362765, 382449, 2145353, 2478785

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
OEIS Wiki, Generalized Fermat numbers

Cf. A152585, A268657, A268658, A204620, A268659, A268661, A268662, A268663, A226366, A268664. Subsequence of A002253.

A268664 Numbers n such that 5*2^n + 1 is a prime factor of a generalized Fermat number 12^(2^m) + 1 for some m.

Original entry on oeis.org

13, 15, 127, 5947, 26607, 1320487

Offset: 1

Arkadiusz Wesolowski, Feb 10 2016

Wilfrid Keller, private communication, 2008.

Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.
C. K. Caldwell, Top Twenty page, Generalized Fermat Divisors (base=12)
OEIS Wiki, Generalized Fermat numbers

Cf. A152585, A268661, A268662, A268663, A226366, A268657, A268658, A204620, A268659, A268660. Subsequence of A002254.

cp's OEIS Frontend

A178426 11 followed by the Fermat numbers A152585.

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

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

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

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

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A199591 Generalized Fermat numbers: 5^(2^n) + 1, n >= 0.

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A199592 Generalized Fermat numbers: 11^(2^n) + 1, n >= 0.

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