A094133 -id:A094133 - cp's OEIS frontend

A276203 Numbers k such that k^9 + 9^k is prime.

2, 76, 122, 422, 2300, 5090, 7166, 58046, 91382, 234178, 314738

Offset: 1

Felix Fröhlich, Aug 27 2016

Numbers k such that A185277(k) is prime.

			2 is a term of the sequence, because A185277(2) = 593 is prime.

Hans Havermann, Indices of A094133 in A076980.
Andrey V. Kulsha, Factorizations of x^y + y^x for 1 < y < x < 151.
P. Leyland, Primes and Strong Pseudoprimes of the form x^y + y^x.

Cf. A064539, A185277, A253471.

PARI
```
is(n) = ispseudoprime(n^9+9^n)
```

A321518 Smallest k > 1 such that n^k + k^n is prime, i.e., a Leyland prime, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 0, 24

Offset: 2

Felix Fröhlich, Nov 12 2018

a(4) = 0. Proof: For k == 1 (mod 4), 4^k + k^4 = 4*x^4 + k^4 = (2*x^2 - 2*k*x + k^2)(2*x^2 + 2*k*x + k^2), where x = 4^((k-1)/4). For k == 3 (mod 4), 4^k + k^4 = 64*x^4 + k^4 = (8*x^2 - 4*k*x + k^2)(8*x^2 + 4*k*x + k^2), where x = 4^((k-3)/4) (cf. Israel, 2015).
Conjecture: a(6) = 0.
From Jon E. Schoenfield, Nov 13 2018: (Start)
Let t = 6^k + k^6.
If k is even, then 2|t.
If k is odd but not divisible by 7, then 7|t.
If k is divisible by 3, then 3|t.
If k == 7 or 63 (mod 70), then 5|t.
Thus, a(6) == 35, 49, 91, 119, 161, or 175 (mod 210) if a(6) > 0. (End)

			For n = 5: 5^24 + 24^5 = 59604644783353249 is prime, and 24 is the smallest k > 1 such that 5^k + k^5 is prime, so a(5) = 24.

Robert Israel, Re: Smallest k > 1 such that n^k+k^n is prime, or 0 if no such k exists, Seqfan (Dec 11 2015).
Wikipedia, Leyland number

Cf. A076980, A094133.

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A276203 Numbers k such that k^9 + 9^k is prime.

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A321518 Smallest k > 1 such that n^k + k^n is prime, i.e., a Leyland prime, or 0 if no such k exists.

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Keywords

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