This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A234309 #19 Dec 25 2013 02:21:57
%S A234309 0,0,0,0,0,1,1,2,2,3,3,4,4,4,4,6,5,6,5,7,7,6,7,7,8,7,7,6,6,7,9,9,6,9,
%T A234309 12,8,6,9,9,9,8,10,8,9,6,9,8,8,10,6,8,11,8,11,8,7,10,8,7,8,7,9,9,11,
%U A234309 11,8,8,9,10,12,7,12,10,8,5,7,9,14,9,9,9,8,7
%N A234309 a(n) = |{2 < k <= n/2: 2^{phi(k)} + 2^{phi(n-k)} - 1 is prime}|, where phi(.) is Euler's totient function.
%C A234309 Conjecture: (i) a(n) > 0 for all n > 5.
%C A234309 (ii) For any integer n > 1, 2^k +2^{phi(n-k)} - 1 is prime for some 0 < k < n, and 2^{sigma(j)} + 2^{phi(n-j)} - 1 is prime for some 0 < j < n, where sigma(j) is the sum of all positive divisors of j.
%C A234309 As phi(k) is even for any k > 2, part (i) of the conjecture implies that there are infinitely many primes of the form 4^a + 4^b - 1 with a and b positive integers (cf. A234310). Note that any Mersenne prime greater than 3 has the form 2^{2a+1} - 1 = 4^a + 4^a - 1.
%H A234309 Zhi-Wei Sun, <a href="/A234309/b234309.txt">Table of n, a(n) for n = 1..8000</a>
%e A234309 a(6) = 1 since 2^{phi(3)} + 2^{phi(3)} - 1 = 2^2 + 2^2 - 1 = 7 is prime.
%e A234309 a(7) = 1 since 2^{phi(3)} + 2^{phi(4)} - 1 = 2^2 + 2^2 - 1 = 7 is prime.
%e A234309 a(8) = 2 since 2^{phi(3)} + 2^{phi(5)} - 1 = 2^2 + 2^4 - 1 = 19 and 2^{phi(4)} + 2^{phi(4)} - 1 = 2^2 + 2^2 - 1 = 7 are both prime.
%t A234309 a[n_]:=Sum[If[PrimeQ[2^(EulerPhi[k])+2^(EulerPhi[n-k])-1],1,0],{k,3,n/2}]
%t A234309 Table[a[n],{n,1,100}]
%Y A234309 Cf. A000010, A000040, A000079, A000668, A233307, A233390, A233542, A233544, A233547, A233566, A233867, A233918, A234200, A234246, A234308, A234310, A234337, A234344, A234346, A234347, A234359, A234360, A234361
%K A234309 nonn
%O A234309 1,8
%A A234309 _Zhi-Wei Sun_, Dec 23 2013