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%I A119768 #19 Jan 05 2020 14:18:01
%S A119768 3,5,17,19,71,73,107,109,881,883,1151,1153,2591,2593,3527,3529,4049,
%T A119768 4051,15137,15139,20807,20809,34847,34849,46817,46819,69191,69193,
%U A119768 83231,83233,103967,103969,112337,112339,139967,139969,149057,149059,176417
%N A119768 Twin prime pairs that sum to a power.
%C A119768 Since twin prime pairs greater than (3,5) occur as either (5,7) mod 12 or (11,1) mod 12, all sums of such twin primes are always divisible by 12. Thus all powers are divisible by 12. The first few terms in base 12 are: 15, 17, 5E, 61, 8E, 91, 615, 617, 7EE, 801, 15EE, 1601 and the corresponding powers are 30, 100, 160, 1030, 1400, 3000.
%H A119768 Amiram Eldar, <a href="/A119768/b119768.txt">Table of n, a(n) for n = 1..10000</a>
%F A119768 If a(n) is the above sequence of twin primes, then a(2n-1),a(2n) is a twin prime pair and a(2n-1)+a(2n) is a power.
%F A119768 a(2*n-1) = A270231(n), a(2*n) = A270231(n) + 2. - _Amiram Eldar_, Jan 05 2020
%e A119768 a(5) + a(6) = 71 + 73 = 144 = 12^2.
%p A119768 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->op(z[1..2]),L);
%t A119768 powQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; aQ[n_] := PrimeQ[n] && PrimeQ[n + 2] && powQ[2 n + 2]; s = Select[Range[10^4], aQ]; Union @ Join[s, s + 2] (* _Amiram Eldar_, Jan 05 2020 *)
%o A119768 (PARI) my(pp=3);forprime(p=5,180000,if(p-pp==2,if(ispower(p+pp),print1(pp,", ",p,", ")));pp=p) \\ _Hugo Pfoertner_, Jan 05 2020
%Y A119768 Cf. A001097, A001359, A006512, A069496, A270231, A330978, A330980.
%K A119768 easy,nonn,tabf
%O A119768 1,1
%A A119768 _Walter Kehowski_, Jun 18 2006
%E A119768 a(1)-a(2) inserted by _Amiram Eldar_, Jan 05 2020

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