A243197 Least number k such that (k+1)^prime(n) - (k-1)^prime(n) is a semiprime.

1, 2, 6, 2, 12, 2, 14, 4, 4, 84, 4, 62, 80, 12, 4, 78, 20, 26, 78, 2, 78, 112, 356, 96, 76, 34, 2, 112, 24, 6, 4, 62, 188, 172, 96, 194, 46, 18, 30, 544, 168, 204, 52, 126, 186, 32, 24, 4, 60, 352, 94, 12, 250, 32, 28, 76, 136, 10, 2794, 4, 46, 16, 178, 58, 886, 104, 74, 54

Offset

1,2

Comments

a(n) is even for n > 1.

(k+1)^M-(k-1)^M is divisible by 2 for any n. Thus, if it is a semiprime, then ((k+1)^M-(k-1)^M)/2 must be prime. This is only prime if M is also prime. Thus we may assume M is a prime.

Links

Table of n, a(n) for n=1..68.

Example

(6+1)^(prime(3))-(6-1)^(prime(3)) = 7^5-5^5 = 13682 = 2*6841. Thus a(3) = 6.

Mathematica

lnk[n_]:=Module[{k=1}, While[PrimeOmega[(k+1)^n-(k-1)^n]!=2, k++]; k]; lnk/@Prime[Range[70]] (* Harvey P. Dale, Jun 05 2015 *)

Prog

(PARI) a(n)=for(k=1, 10^4, if(ispseudoprime(((k+1)^prime(n)-(k-1)^prime(n))/2), return(k)))
n=1; while(n<100, print1(a(n), ", "); n+=1)

Crossrefs

Sequence in context: A136698 A223473 A292522 * A319048 A066900 A356016

Adjacent sequences: A243194 A243195 A243196 * A243198 A243199 A243200

Keyword

nonn

Author

Derek Orr, Jun 01 2014

Status

approved