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A309120 a(n) is the least k > 1 such that n*k is adjacent to a prime.

%I A309120 #39 Jul 19 2019 17:37:02
%S A309120 2,2,2,2,2,2,2,2,2,2,2,2,4,2,2,2,4,2,2,2,2,2,2,2,4,2,2,3,2,2,2,3,2,2,
%T A309120 2,2,2,3,2,2,2,2,4,2,2,3,6,2,2,2,2,2,2,2,2,2,2,3,6,3,6,5,2,2,2,2,4,2,
%U A309120 2,2,4,5,4,2,2,2,4,2,2,3,2,2,2,2,6,2,2,3,2,2,2,3,4,3
%N A309120 a(n) is the least k > 1 such that n*k is adjacent to a prime.
%C A309120 If n is odd then a(n) is even.
%C A309120 a(n) exists by Dirichlet's theorem on primes in arithmetic progressions.
%H A309120 Robert Israel, <a href="/A309120/b309120.txt">Table of n, a(n) for n = 1..10000</a>
%F A309120 a(A104278(n)) > 2 and a(A147820(n)) = 2. - _Ivan N. Ianakiev_, Jul 18 2019
%e A309120 a(13)=4 because 4*13+1=53 is prime but none of 2*13-1,2*13+1,3*13-1,3*13+1 are primes.
%p A309120 f:= proc(m) local k;
%p A309120   for k from 2 by 1+(m mod 2) do
%p A309120     if isprime(k*m-1) or isprime(k*m+1) then return k fi
%p A309120   od
%p A309120 end proc:
%p A309120 map(f, [$1..100]);
%t A309120 a[n_]:=Module[{k=2},While[Not[PrimeQ[k*n-1]||PrimeQ[k*n+1]],k++];k];
%t A309120 a/@Range[94] (* _Ivan N. Ianakiev_, Jul 18 2019 *)
%o A309120 (PARI) a(n) = my(k=2); while (!isprime(n*k+1) && !isprime(n*k-1), k++); k; \\ _Michel Marcus_, Jul 19 2019
%Y A309120 Cf. A307833, A104278, A147820.
%K A309120 nonn
%O A309120 1,1
%A A309120 _Robert Israel_, Jul 17 2019