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A119767 Perfect powers which are the sum of twin prime pairs.

%I A119767 #26 Sep 18 2024 16:33:34
%S A119767 8,36,144,216,1764,2304,5184,7056,8100,30276,41616,69696,93636,138384,
%T A119767 166464,207936,224676,279936,298116,352836,360000,412164,562500,
%U A119767 725904,777924,876096,944784,956484,1077444,1299600,1468944,1617984,1920996,2160900,2286144,2304324,2509056
%N A119767 Perfect powers which are the sum of twin prime pairs.
%C A119767 Since twin primes greater than (3,5) are either occur as (5,7) mod 12 or (11,1) mod 12, all sums of such twin primes are divisible by 12. Thus all powers are divisible by 12 and are best looked at in base 12. For example, a(3) = 5E + 61 = 100, where E is eleven.
%H A119767 Jens Kruse Andersen, <a href="/A119767/b119767.txt">Table of n, a(n) for n = 1..10000</a>
%e A119767 8 = 2^3 = 3 + 5 (twin primes). Thus 8 is a member of this sequence.
%e A119767 36 = 6^2 = 17 + 19 (twin primes). Thus 36 is a member of this sequence.
%e A119767 a(3) = 71 + 73 = 144.
%p A119767 egcd := proc(n::nonnegint) local L; if n=0 or n=1 then n else L:=ifactors(n)[2]; L:=map(z->z[2],L); igcd(op(L)) fi end: L:=[]: for w to 1 do for x from 1 to 2*12^2 do s:=6*x; for r from 2 to 79 do t:=s^r; if egcd(s)=1 and andmap(isprime,[(t-2)/2,(t+2)/2]) then print((t-2)/2,(t+2)/2,t)); L:=[op(L),[(t-2)/2,(t+2)/2,t]]; fi; od od od; L:=sort(L,(a,b)->a[1]<b[1]); map(z->z[3],L);
%t A119767 Lim=2600000;ts=Select[Prime[Range[PrimePi[Lim]]], PrimeQ[# + 2] &]2+2;pp=Join[{1}, Select[Range[Lim], GCD@@FactorInteger[#][[All, 2]]>1&]] ;s={};Do[ If[MemberQ[ pp,ts[[n]]],AppendTo[s,ts[[n]]]] ,{n,Length[ts]}];s (* _James C. McMahon_, Sep 18 2024 *)
%o A119767 (PARI) a(N) = for(n=1,N,if(ispower(n),if(nextprime(n/2)-precprime(n/2)==2&&precprime(n/2)+nextprime(n/2)==n,print1(n,", ")))) \\ vary the program's range for any N; _Derek Orr_, Jul 27 2014
%Y A119767 Cf. A037072, A245591.
%K A119767 easy,nonn
%O A119767 1,1
%A A119767 _Walter Kehowski_, Jun 18 2006
%E A119767 _R. J. Mathar_ pointed out that 8 was missing. Once corrected, the old A245591 could be merged into this entry. - _N. J. A. Sloane_, Jul 30 2014