A122902 -id:A122902 - cp's OEIS frontend

A080121 a(n) is the smallest k > 0 such that n^2^k + (n+1)^2^k is prime, or -1 if no such k exists.

1, 1, 2, 1, 1, 2, 1, 2, 1, 5

Offset: 1

T. D. Noe, Jan 29 2003

This sequence is the base-2 logarithm of A077659. It is known that a(11) > 22. Is it possible that 11^2^k + 12^2^k is composite for all k > 0?

The corresponding primes are listed in A122900. Currently a(n) is unknown for n in {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 0 < k < 10 are checked. The first occurrence of each exponent k is listed in A122902. - Alexander Adamchuk, Sep 18 2006

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

Cf. A058064, A057856, A077659, A122900, A122902.

If A058064(n) > 0, then a(n) = A058064(n). - Max Alekseyev, Sep 10 2020

Edited by Max Alekseyev, Sep 09 2020

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

cp's OEIS Frontend

A080121 a(n) is the smallest k > 0 such that n^2^k + (n+1)^2^k is prime, or -1 if no such k exists.

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