OFFSET
2,1
COMMENTS
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.
The initial records in this sequence are a(2)=6, a(3)=7, a(2055)=8. No higher value was found up to 5500000. It is not clear whether a(n) is bounded.
Heuristically, Prob(a(n) > m) ~ (2/log n)^m/m! as n -> infinity for fixed m. The sum over n diverges, so we should expect infinitely many a(n) > m. - Robert Israel, Aug 12 2014
a(215539779) = 9 is a record and there is no higher value up to 4*10^9. a(n) <= 3 for all even n > 2, since n-1 divides n^3-1 and n+1 divides n^3+1. - Jens Kruse Andersen, Aug 14 2014
LINKS
Stanislav Sykora, Table of n, a(n) for n = 2..10000
EXAMPLE
a(4)=3 because 4^1 and 4^2 are bracketed by the odd numbers (3,5) and (15,17) and each pair contains a prime, but 4^3 is bracketed by (63,65) which are both nonprimes.
a(5)=4 because 5^1, 5^2, and 5^3 are bracketed by odd pairs (3,7), (23,27) and (123,127) which all contain at least one prime. But 5^4 is bracketed by odd numbers (623,627) which are both composites.
MAPLE
f:= proc(n) local m, nm;
for m from 1 do
nm:= n^m;
if n::odd then if not isprime(nm+2) and not isprime(nm-2) then return(m) fi
elif not isprime(nm+1) and not isprime(nm-1) then return(m)
fi
od
end proc:
seq(f(n), n=2..1000); # Robert Israel, Aug 12 2014
MATHEMATICA
a245513Q[n_Integer] := Module[{i},
Catch[For[i = 0, i <= 20, i++,
If[EvenQ[n],
If[! PrimeQ[n^i + 1] && ! PrimeQ[n^i - 1], Throw[i]],
If[! PrimeQ[n^i + 2] && ! PrimeQ[n^i - 2], Throw[i]]
]]]]; a245513[n_Integer] := a245513Q /@ Range[2, n]; a245513[120] (* Michael De Vlieger, Aug 12 2014 *)
PROG
(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); }
a=avector(10000) \\ For nmax=6000000 runs out of 1GB memory
CROSSREFS
KEYWORD
nonn
AUTHOR
Stanislav Sykora, Jul 24 2014
STATUS
approved