cp's OEIS Frontend

A239711 Twin primes of the form m = b^i + b^j +- 1, where i > j > 0, b > 1.

%I A239711 #15 Oct 26 2014 05:00:10
%S A239711 5,7,11,13,17,19,29,31,41,43,71,73,107,109,149,151,191,193,239,241,
%T A239711 269,271,419,421,461,463,599,601,809,811,1031,1033,1151,1153,1301,
%U A239711 1303,1451,1453,1481,1483,1721,1723,1871,1873,2111,2113,2267,2269,2549,2551,2969,2971,3389,3391,3539,3541
%N A239711 Twin primes of the form m = b^i + b^j +- 1, where i > j > 0, b > 1.
%C A239711 (a(2k-1), a(2k)), k > 0, form pairs of twin primes.
%C A239711 Numbers m that satisfy m = b^i + b^j + 1 and b == 1 (mod 3) and those that satisfy m = b^i + b^j - 1 with odd i and j and b == 2 (mod 3) are never terms, since they are divisible by 3. It follows that no numbers 4^i + 4^j +- 1, or 7^i + 7^j +- 1, or 10^i + 10^j +- 1, ... can be terms. Also, no numbers 5^(2m-1) + 5^(2k-1) +- 1, or 8^(2m-1) + 8^(2k-1) +- 1, or 11^(2m-1) + 11^(2k-1) +- 1, ... with m > k > 0, can be terms.
%C A239711 Example 1: 10^6 + 10^4 + 1 = 1010001 is not a term, since 10 == 1 (mod 3); certainly, 1010001 = 3*336667.
%C A239711 Example 2: 8^9 + 8^7 - 1 = 136314879 is not a term, since 8 == 2 (mod 3) and i, j odd; certainly 136314879 = 3*45438293.
%H A239711 Hieronymus Fischer, <a href="/A239711/b239711.txt">Table of n, a(n) for n = 1..10000</a>
%e A239711 a(1) = 5, since 5 = 2^2 + 2^1 - 1 is prime.
%e A239711 a(2) = 7, since 7 = 2^3 + 2^1 + 1 is prime.
%e A239711 a(7) = 29, since 29 = 3^3 + 3^1 - 1 is prime.
%e A239711 a(8) = 31, since 31 = 3^3 + 3^1 + 1 is prime.
%e A239711 a(9) = 41.
%e A239711 a(10) = 43.
%e A239711 a(99) = 43889.
%e A239711 a(100) = 43891.
%e A239711 a(999) = 233524241.
%e A239711 a(1000) = 233524243.
%e A239711 a(9999) = 110211052379.
%e A239711 a(10000) = 110211052381.
%e A239711 a(99999) = 27208914574871.
%e A239711 a(100000) = 27208914574873.
%e A239711 a(199999) = 136140088764371.
%e A239711 a(200000) = 136140088764373.
%e A239711 [the last two terms form the 100000th twin prime pair of the form b^i + b^j +-1]
%Y A239711 Cf. A239709, A239710, A239712 - A239720.
%K A239711 nonn
%O A239711 1,1
%A A239711 _Hieronymus Fischer_, Mar 27 2014 and May 04 2014