A097501 p^q + q^p for twin primes p and q.

368, 94932, 36314872537968, 244552822542936127033092, 2177185942561672462146321298650240665136431700, 2246585380039521951243337580678537047744572047581514711375688196554564

Offset: 2

Author

Cino Hilliard, Aug 25 2004

Keywords

• nonn

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)?

Examples

			Consider the second twin prime pair (5,7). 5^7 + 7^5 = 94932, the 2nd entry.
		

Crossrefs

Cf. A051442.

Programs

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 *)

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",")))