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 A233466 #64 Oct 30 2023 13:03:48
%S A233466 165,64005,6992962170388485,18446744047939747845
%N A233466 Numbers k such that phi(k) = (k-5)/2.
%C A233466 According to the following theorem I discovered recently, the 20-digit number 18446744047939747845 is in the sequence.
%C A233466 Theorem: If k and m are integers, k < 6, and p = 2^2^k + m is a prime such that p does not divide 2^2^k-1, then p*(2^2^k-1) is a solution to the equation phi(x) = (x+m)/2.
%C A233466 Note that m cannot be -1 and for k < 6, 2^2^k-1 is the product of the first k Fermat primes.
%C A233466 Take m=-5; since 2^2^2-5, 2^2^3-5 and 2^2^5-5 are prime we get three terms of the sequence.
%C A233466 Take m=1; since 2^2^0+1, 2^2^1+1, 2^2^2+1, 2^2^3+1 and 2^2^4+1 are prime (Fermat primes) we get five terms of the sequence A050474.
%C A233466 Conjecture (i): There is no solution to the equation phi(x) = (x-1)/2.
%C A233466 Conjecture (ii): The sequence has only three terms and a(3) = (2^2^5-5) * (2^2^5-1) = 18446744047939747845.
%C A233466 Conjecture (i) is a part of Lehmer's totient problem. Conjecture (ii) is disproved with the term a(3) = 6992962170388485 = 3 * 5 * 17 * 353 * 929 * 83623931. - _Max Alekseyev_, Oct 28 2023
%C A233466 a(5) <= 202317618492499837497376768005 = 3 * 5 * 17 * 257 * 65951 * 10414721 * 4494603392933. - _Max Alekseyev_, Oct 30 2023
%H A233466 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer%27s_totient_problem">Lehmer's totient problem</a>.
%e A233466 phi(165) = 80 = (165-5)/2.
%t A233466 Do[If[EulerPhi[n]==1/2(n-5),Print[n]],{n,1,70001,4}]
%o A233466 (PARI) is(n)=eulerphi(n)==(n-5)/2 \\ _Charles R Greathouse IV_, Jan 20 2014
%o A233466 (Python)
%o A233466 from itertools import islice, count
%o A233466 from sympy import totient
%o A233466 def A233466gen(): return filter(lambda n:2*totient(n) == n-5,count(1,2))
%o A233466 A233466_list = list(islice(A233466gen(),2)) # _Chai Wah Wu_, Dec 15 2021
%Y A233466 Subsequence of A083255. - _R. J. Mathar_, Jan 13 2014
%Y A233466 Cf. A000010, A050474.
%K A233466 nonn,more,hard,nice
%O A233466 1,1
%A A233466 _Farideh Firoozbakht_, Dec 26 2013
%E A233466 Missing term a(3) inserted, a(4) confirmed by _Max Alekseyev_, Oct 28 2023