A123599 -id:A123599 - cp's OEIS frontend

A259835 a(n) is the number of odd primes of the form b^(2^n)+1 that are less than A123599(n+1).

1, 1, 1, 1, 41, 152, 122, 185, 8, 860, 24, 612, 97094

Offset: 0

Jeppe Stig Nielsen, Jul 06 2015

A generalized Fermat prime b^(2^n)+1 can be thought of as belonging to the "family" n. Then a(n) counts how many generalized Fermat primes in family n precede the first generalized Fermat prime in family n+1.
Each family as defined here is a subset of its preceding family, in the sense that b^(2^n) + 1 = (b^2)^(2^(n-1)) + 1.
a(12) is expected to be near 97000.

			To find a(5), find all primes b^32 + 1 until you reach a base b that is a perfect square. In this case you find 152 nonsquare b values { 30, 54, 96, 112, ..., 10396 }, but the 153rd b is 10404, a perfect square. So 10404^32 + 1 = 102^64 + 1 belongs to the next family. Therefore a(5)=152.

Y. Gallot, Status of the smallest base values yielding Generalized Fermat primes.

Cf. A056993, A123599.

b=2;for(n=0,100,x=0;until(,if(ispseudoprime(b^(2^n)+1),if(issquare(b,&b),break,x++));b+=2);print("a(",n,")=",x,", next b is ",b))

a(12) via b-file of A088362 from Jeppe Stig Nielsen, Feb 16 2022

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

A111635 Smallest prime of the form x^(2^n) + y^(2^n) where x,y are distinct integers.

Original entry on oeis.org

2, 5, 17, 257, 65537, 3512911982806776822251393039617, 4457915690803004131256192897205630962697827851093882159977969339137, 1638935311392320153195136107636665419978585455388636669548298482694235538906271958706896595665141002450684974003603106305516970574177405212679151205373697500164072550932748470956551681

Offset: 0

Max Alekseyev, Aug 09 2005

nonn

Is this sequence defined for all n?
From Jeppe Stig Nielsen, Sep 16 2015: (Start)
Numbers of this form are sometimes called extended generalized Fermat numbers.
If we restrict ourselves to the case y=1, we get instead the sequence A123599, therefore a(n) <= A123599(n) for all n. Can this be an equality for some n > 4?
The formula x^(2^m) + y^(2^m) also gives the decreasing chain {A000040, A002313, A002645, A006686, A100266, A100267, ...} of subsets of the prime numbers if we drop the requirement that x != y and take all primes (not just the smallest one) with m greater than some lower bound.
(End)
For more terms (the values of max(x,y)), see A291944. - Jeppe Stig Nielsen, Dec 28 2019

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..9

Cf. A019434, A100270, A123599, A291944.

A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

N. J. A. Sloane, Table of n, a(n) for n = 1..6

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Edited by R. J. Mathar, Aug 22 2009

Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A164312 Numbers n such that k^n + (k-1)^n + ... + 3^n + 2^n + 1 is prime for some k.

Original entry on oeis.org

1, 2, 4, 8, 16, 1440

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

These terms have k-values {2, 2, 2, 2, 2, 5} respectively. When k = 2, the prime mentioned in the definition is given in A164307. - Derek Orr, Jun 06 2014

			1^1 + 2^1 = 3 is prime (k = 2).
1^2 + 2^2 = 5 is prime (k = 2).
1^4 + 2^4 = 17 is prime (k = 2).
1^8 + 2^8 = 257 is prime (k = 2).
1^16 + 2^16 = 65537 is prime (k = 2).
1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379 is prime (k = 5).

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599, A164307.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,x];Print[Date[],x]],{n,4!}],{x,7!}];lst

a(n)=for(k=1,10^3,if(ispseudoprime(sum(i=1,k,i^n)),return(k)))
n=1;while(n<5000,if(a(n),print1(n,", "));n++) \\ Derek Orr, Jun 06 2014

Definition improved by Derek Orr, Jun 06 2014

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A259835 a(n) is the number of odd primes of the form b^(2^n)+1 that are less than A123599(n+1).

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

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A111635 Smallest prime of the form x^(2^n) + y^(2^n) where x,y are distinct integers.

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A164307 Primes in A081175.

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A164312 Numbers n such that k^n + (k-1)^n + ... + 3^n + 2^n + 1 is prime for some k.

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