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a(n) is the least prime factor of 2^n-n-2.
1

%I #13 Nov 23 2022 08:57:32

%S 3,2,5,2,7,2,3,2,5,2,13,2,3,2,13,2,19,2,3,2,7,2,5,2,3,2,29,2,5,2,3,2,

%T 73,2,23,2,3,2,29,2,43,2,3,2,47,2,7,2,3,2,37,2,113,2,3,2,11,2,61,2,3,

%U 2,5,2,67,2,3,2,5,2,73,2,3,2,53,2,79,2,3,2,11,2,5,2,3,2,61,2,5,2,3,2

%N a(n) is the least prime factor of 2^n-n-2.

%C a(n) = 2 if n is even.

%C a(n) = 3 if n == 3 (mod 6).

%C a(n) = 5 if n == 5 or 11 (mod 20) and is not divisible by 3.

%C a(n) <= n if n is prime.

%C a(n) = A000247(n) for n = 3 and (subject to confirmation of probable primes) 39137 and 59819. The latter two were discovered by Henri Lifchitz in 2005.

%H Robert Israel, <a href="/A358536/b358536.txt">Table of n, a(n) for n = 3..516</a>

%H R. Israel and R. Fernando, <a href="https://math.stackexchange.com/questions/4581358/primes-2n-n-2">Primes 2^n-n-2</a>, Mathematics StackExchange (2022).

%F a(n) = A020639(A000247(n)).

%e a(5) = 5 because 2^5 - 5 - 2 = 25 has least prime factor 5.

%p f:= proc(n) local F;

%p F:= select(type,map(t -> t[1],ifactors(2^n-n-2,easy)[2]),posint);

%p if F = [] then F:= map(t -> t[1], ifactors(2^n-n-2)[2])) fi;

%p min(F);

%p end proc:

%p map(f, [$3..100]);

%Y Cf. A000247, A020639.

%K nonn

%O 3,1

%A _J. M. Bergot_ and _Robert Israel_, Nov 21 2022