A132260 -id:A132260 - cp's OEIS frontend

A132261 Main diagonal of array in A132260.

13, 107, 101, 491, 8039, 9349

Offset: 1

Jonathan Vos Post, Aug 15 2007

			a(1) = 13 because 13 is the first prime p such that 2^2^3 + p^2^3 is prime.
a(2) = 107 because 107 is the 2nd prime p such that 2^2^4 + p^2^4 is prime.
a(3) = 101 because 101 is the 3rd prime p such that 2^2^5 + p^2^5 is prime.

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132215, A132216, A132260.

a(n) = A[n,n+2] = n-th prime p such that 2^2^(n+2) + p^2^(n+2) is prime.

A375217 Primes p such that p^64 + 2^64 is prime.

Original entry on oeis.org

37, 53, 181, 491, 547, 619, 661, 677, 911, 941, 1297, 1423, 1867, 2441, 2687, 3137, 3571, 5387, 5821, 5881, 6449, 6551, 6899, 8263, 8537, 8999, 9803, 9931, 10861, 11057, 11131, 11423, 12377, 12941, 13147, 14009, 14519, 14759, 14813, 15493, 16103, 16573, 19949

Offset: 1

Mykhailo Papenko, Oct 17 2024

nonn

It is conjectured that solutions for p1^n + p2^n = p3 (where p1, p2, and p3 are all primes and n is a natural number) exist only when n is itself a power of two (when n is a number in A000079); and would have infinitely many solutions.

But it's proven that either p1 or p2 must be a 2.

Mykhailo Papenko, Table of n, a(n) for n = 1..8079
Mykhailo Papenko, Primes-Made-Up-of-Primes, Github.

6th row of A132260.

/* see link for code with instructions */

Select[Prime[Range[2255]],PrimeQ[#^64+2^64]&] (* James C. McMahon, Nov 19 2024 *)

p^64 + 2^64 ∈ A000040 (p^64 + 2^64 belongs to the set of the prime numbers)

A378490 Least prime p such that p^(2^n) + 2^(2^n) is prime.

Original entry on oeis.org

3, 3, 3, 13, 89, 29, 37, 113, 113, 13, 1151, 43, 53, 5503

Offset: 0

Jean-Marc Rebert, Nov 28 2024

			a(1) = 3, because 3^(2^1) + 2^(2^1) = 13 is prime and no smaller prime satisfies the condition.

Cf. A001146, A132260, A157764, A375215.

a(n) = my(p=2, q=2^n); while (!ispseudoprime(p^q + 2^q), p=nextprime(p+1)); p; \\ Michel Marcus, Nov 28 2024

a(n) = A132260(n,1) for n>=3.

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A132261 Main diagonal of array in A132260.

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A375217 Primes p such that p^64 + 2^64 is prime.

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A378490 Least prime p such that p^(2^n) + 2^(2^n) is prime.

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