OFFSET
1,1
COMMENTS
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
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.
EXAMPLE
a(5) + a(6) = 71 + 73 = 144 = 12^2.
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]<b[1]); map(z->op(z[1..2]), L);
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Walter Kehowski, Jun 18 2006
EXTENSIONS
a(1)-a(2) inserted by Amiram Eldar, Jan 05 2020
STATUS
approved