|
%I #19 Jan 05 2020 14:18:01
%S 3,5,17,19,71,73,107,109,881,883,1151,1153,2591,2593,3527,3529,4049,
%T 4051,15137,15139,20807,20809,34847,34849,46817,46819,69191,69193,
%U 83231,83233,103967,103969,112337,112339,139967,139969,149057,149059,176417
%N Twin prime pairs that sum to a power.
%C 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 Amiram Eldar, <a href="/A119768/b119768.txt">Table of n, a(n) for n = 1..10000</a>
%F 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 a(2*n-1) = A270231(n), a(2*n) = A270231(n) + 2. - _Amiram Eldar_, Jan 05 2020
%e a(5) + a(6) = 71 + 73 = 144 = 12^2.
%p 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 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 (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 Cf. A001097, A001359, A006512, A069496, A270231, A330978, A330980.
%K easy,nonn,tabf
%O 1,1
%A _Walter Kehowski_, Jun 18 2006
%E a(1)-a(2) inserted by _Amiram Eldar_, Jan 05 2020
|
