A245513 Smallest m such that neither of the two odd numbers that bracket n^m is a prime.

6, 7, 3, 4, 3, 3, 2, 6, 3, 2, 2, 3, 3, 6, 3, 2, 2, 4, 3, 3, 2, 1, 3, 2, 1, 4, 2, 5, 2, 2, 2, 3, 1, 3, 3, 1, 2, 3, 3, 2, 2, 3, 2, 5, 2, 1, 2, 3, 1, 2, 2, 1, 3, 3, 1, 3, 2, 2, 2, 3, 2, 6, 1, 2, 3, 1, 2, 5, 2, 4, 2, 2, 3, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 2, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 2

Offset: 2

Author

Stanislav Sykora, Jul 24 2014

Keywords

• nonn

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

Examples

			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.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A245509, A245510, A245511, A245512, A245514.

Programs

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 *)

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