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 A350142 #16 May 12 2025 19:27:44 %S A350142 3,5,17,65,257,65537,4294967297 %N A350142 Numbers m of the form 2^k + 1 such that tau(m-2) = tau(m-1) - 1. %C A350142 Corresponding pairs of values [tau(m-2), tau(m-1)]: [1, 2], [2, 3], [4, 5], [6, 7], [8, 9], [16, 17], [32, 33], ... %C A350142 There are no other terms <= 2^1206 + 1 (from A046801 data). %C A350142 The first 5 known Fermat primes from A019434 are in this sequence. Corresponding values of tau(A019434(n - 2)): 1, 2, 4, 8, 16, ... %C A350142 Conjecture 1: Also numbers m of the form 2^k + 1 such that tau(m - 2) = k. %C A350142 Conjecture 2: If 6th Fermat prime F_p6 exists, then tau(F_p6 - 2) is a power of 2 and tau(F_p6 - 1) = tau(F_p6 - 2) + 1. %C A350142 Conjecture 3: Sequence is finite with 7 terms; supersequence of A262534. %e A350142 For number 257 holds: tau(255) = 8, tau(256) = 9. %o A350142 (Magma) [2^k + 1: k in [1..50] | #Divisors(2^k) - #Divisors(2^k-1) eq 1]; %Y A350142 Cf. A000005, A019434, A262534, A343144, A347078. %Y A350142 Intersection of (A055927+2) and A000051. %K A350142 nonn,hard,more %O A350142 1,1 %A A350142 _Jaroslav Krizek_, Dec 16 2021