OFFSET
2,1
COMMENTS
Except for the first term, 6 divides a(n). Let p = 3k+2 for odd k since k even implies p even, a contradiction. Then p = 6m + 5 and q = 6m+7 = 6m1 + 1. So p^q+q^p = (6m+5)^(6m1+1) + (6m1+1)^(6m+5) = 6H + 5^odd + 1^odd. Now 5 = (6-1) and (6-1)^odd + 1 = 6G -1 + 1 = 6G as stated. Are 3 and 17 the only primes in A051442(n)?
EXAMPLE
Consider the second twin prime pair (5,7). 5^7 + 7^5 = 94932, the 2nd entry.
MATHEMATICA
lst={}; Do[p=Prime[n]; If[PrimeQ[q=p+2], a=(p^q+q^p); AppendTo[lst, a]], {n, 2*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)
#[[1]]^#[[2]]+#[[2]]^#[[1]]&/@Select[Partition[Prime[Range[20]], 2, 1], #[[2]] - #[[1]]==2&] (* Harvey P. Dale, Sep 07 2019 *)
PROG
(PARI) f(n) = for(x=1, n, p=prime(x); q=prime(x+1); if(q-p==2, v=p^q+q^p; print1(v", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Cino Hilliard, Aug 25 2004
STATUS
approved