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

368, 94932, 36314872537968, 244552822542936127033092, 2177185942561672462146321298650240665136431700, 2246585380039521951243337580678537047744572047581514711375688196554564

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)?

Links

Table of n, a(n) for n=2..7.

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

Cf. A051442.

Sequence in context: A173055 A318938 A240006 * A239411 A239339 A352224

Adjacent sequences: A097498 A097499 A097500 * A097502 A097503 A097504

Keyword

nonn

Author

Cino Hilliard, Aug 25 2004

Status

approved