A080121 -id:A080121 - cp's OEIS frontend

A122902 First occurrence of exponent n in A080121 corresponding to the minimum prime of the form (k^(2^n) + (k+1)^(2^n)) = A122900(k).

1, 3, 23, 21, 10, 95, 255, 86, 59

Offset: 1

Alexander Adamchuk, Sep 18 2006, Oct 01 2006

Minimum primes of the form n^(2^m) + (n+1)^(2^m) are listed in A122900. The exponents m are listed in A080121.

a(10)-a(13)>1000, a(14)-a(16)>100.

			A080121 begins with 1,1,2,1,1,2,1,2,1,5,?,1,2,1,?,2,1,?,1,?,4,1,3,1,..., where the unknown terms (denoted with ?) are at least 10. So a(1) = 1, a(2) = 3, a(3) = 23, a(4) = 21, a(5) = 10.

T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A077659, A080121, A122900, A080208, A078902, A080134, A019434, A153504, A152913, A194185.

Edited by Max Alekseyev, Sep 09 2020

A122901 Erroneous duplicate of A080121.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 1, 5, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 4, 1, 3, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 4, 1, 2, 0, 4, 0, 1, 2, 0, 1, 0, 0, 0, 2, 0, 4, 0, 0, 9, 1, 0, 0, 2, 0, 1, 3, 2, 2, 1, 1, 0, 1, 0, 2, 4, 3, 0, 2, 1, 4, 0, 1, 0, 1, 1, 8, 1, 2, 2, 1, 0, 0, 4, 0, 6, 4, 1, 2, 1, 1

Offset: 1

Alexander Adamchuk, Sep 18 2006

dead

A077659 a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 2, 4, 2, 2, 4, 2, 4, 2, 32

Offset: 1

T. D. Noe, Nov 14 2002

Checking k up through 1024 suggests that the sequence may continue -1, 2, 4, 2, -1, 4, 2, -1, 2, -1, 16, 2, 8, 2, 2, 4, 4, -1, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, ...
For any a>1 and b>1, a^k + b^k is composite for all odd k>1. Hence if n^k + (n+1)^k is prime then k must be a power of 2.
It is known that a(11) > 2^22. Is it possible that 11^2^m + 12^2^m is composite for all m > 0?

			a(3)=4 because 3^2 + 4^2 = 25 is not prime, but 3^4 + 4^4 = 337 is prime.

T. D. Noe, Factorizations of Generalized Fermat Numbers 12^2^k + 11^2^k

Cf. A078902.

Cf. A080121.

lst={}; For[n=1, n<=100, n++, k=2; While[k<=2^10 && !PrimeQ[n^k+(n+1)^k], k=2*k]; If[k<=2^10, AppendTo[lst, k], AppendTo[lst, -1]]]; lst

A122900 Minimum prime of the form n^k + (n+1)^k for k>1, or 0 if no such prime exists.

Original entry on oeis.org

5, 13, 337, 41, 61, 3697, 113, 10657, 181, 2211377674535255285545615254209921

Offset: 1

Alexander Adamchuk, Sep 18 2006

Currently a(n) is unknown for n = {11, 15, 18, 20, 28, 44, 46, 49, 51, 52, 53, 55, 57, 58, 61, 62, 64, 71, 73, 77, 81, 83, 91, 92, 94, ...}. All n < 100 and 1 < k < 2^10 have been checked.
All nonzero a(n) have a form n^(2^m) + (n+1)^(2^m).
The exponents m are listed in A080121. The first occurrence of each exponent m in A080121 is listed in A122902.

			a(1) = 5 because 1^2 + 2^2 = 5 is prime.
a(2) = 13 because 2^2 + 3^2 = 13 is prime.
a(3) = 337 because 3^4 + 4^4 = 337 is prime but 3^3 + 4^3 = 91 and 3^2 + 4^2 = 25 are composite.

Cf. A077659, A080121, A122902, A080208.

Edited by Max Alekseyev, Sep 09 2020

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

A122902 First occurrence of exponent n in A080121 corresponding to the minimum prime of the form (k^(2^n) + (k+1)^(2^n)) = A122900(k).

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A122901 Erroneous duplicate of A080121.

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A077659 a(n) = smallest k>1 such that the sum n^k + (n+1)^k is prime, or -1 if no such k exists.

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A122900 Minimum prime of the form n^k + (n+1)^k for k>1, or 0 if no such prime exists.

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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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