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

8, 36, 144, 216, 1764, 2304, 5184, 7056, 8100, 30276, 41616, 69696, 93636, 138384, 166464, 207936, 224676, 279936, 298116, 352836, 360000, 412164, 562500, 725904, 777924, 876096, 944784, 956484, 1077444, 1299600, 1468944, 1617984, 1920996, 2160900, 2286144, 2304324, 2509056

Offset: 1

Author

Walter Kehowski, Jun 18 2006

Keywords

• easy
• nonn

Comments

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.

Examples

			8 = 2^3 = 3 + 5 (twin primes). Thus 8 is a member of this sequence.
36 = 6^2 = 17 + 19 (twin primes). Thus 36 is a member of this sequence.
a(3) = 71 + 73 = 144.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A037072, A245591.

Programs

Maple

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]z[3],L);

Mathematica

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 *)

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

Extensions

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