A389883 a(n) is the smallest integer k such that ((2^n + 5)^k - 5^k)/2^n is prime, or -1 if no such k exists.

2, 3, 3, 5, 3, 3, 7, 3, 173, 5, 199, 3, 179, 3, 227, 7

Offset

0,1

Comments

From Stefano Spezia, Nov 19 2025: (Start)

Does a(n) exist for all n? Is a(n) prime for all n?

Some more terms: a(17) = 7, a(18) = 5, a(20) = 13, a(23) = 3, a(24) = 79, a(26) = 3, a(27) = 53, a(33) = 181. (End)

a(n) is prime for all n since for each n the expression is of the form ((c+d)^k - d^k)/c, which is not prime if k is not prime. - Michael S. Branicky, Nov 25 2025

Links

Table of n, a(n) for n=0..15.

Example

a(0) = 2 because ((2^0 + 5)^2 - 5^2)/2^0 = 11 is prime.

Mathematica

a[n_]:=Module[{k=0}, While[!PrimeQ[((2^n+5)^k-5^k)/2^n], k++]; k]; Array[a, 17, 0] (* Stefano Spezia, Nov 19 2025 *)

Prog

(Magma) [Min([k: k in  [1..200] | IsPrime(((2^n+5)^k-5^k) div 2^n)]): n in [0..9]];

Crossrefs

Cf. A062572, A128344, A128346, A128349, A246005, A389827.

Sequence in context: A381926 A131320 A020483 * A119912 A076368 A279931

Adjacent sequences: A389880 A389881 A389882 * A389884 A389885 A389886

Keyword

nonn,more

Author

Juri-Stepan Gerasimov, Nov 19 2025

Extensions

a(15) from Stefano Spezia, Nov 19 2025

Status

approved