A245514 - OEIS

%I #12 Feb 03 2018 12:27:20

%S 1,1,2,2,2,1,1,3,1,1,2,1,1,3,1,1,2,1,1,3,1,1,1,1,1,1,1,1,2,1,1,1,1,1,

%T 1,1,1,2,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,

%U 1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N Smallest m such that at least one of the two odd numbers which bracket n^m is not a prime.

%C The locution "the two odd numbers which bracket n^m" indicates the pair (n^m-1,n^m+1) for even n and (n^m-2,n^m+2) for odd n.

%C The initial records in this sequence are a(2)=1, a(4)=2, a(9)=3, a(102795)=4. No higher value was found up to 5500000. It is not clear whether a(n) is bounded.

%H Stanislav Sykora, <a href="/A245514/b245514.txt">Table of n, a(n) for n = 2..10000</a>

%e a(2)=1 because one of the two odd numbers (1,3) which bracket 2^1 is not a prime. a(5)=2 because 5^1 is bracketed by the odd numbers (3,7) which are both prime, while 5^2 is bracketed by the odd numbers (23,27), one of which is not a prime.

%e The number c=102795 is the smallest one whose powers c^1, c^2, c^3 are all odd-bracketed by primes, while c^4 is not.

%o (PARI) avector(nmax)={my(n, k, d=2, v=vector(nmax)); for(n=2, #v+1, d=3-d; k=1; while(1, if((!isprime(n^k-d))||(!isprime(n^k+d)), v[n-1]=k; break, k++)); ); return(v); }

%o a=avector(10000)  \\ For nmax=6000000 runs out of 1GB memory

%Y Cf. A245509, A245510, A245511, A245512, A245513.

%K nonn

%O 2,3

%A _Stanislav Sykora_, Jul 24 2014