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 A325116 #16 Apr 06 2019 04:11:01 %S A325116 0,1,1,3,1,2,1,3,1,1,1,7,1,1,1,3,1,2,1,3,1,1,1 %N A325116 Length of longest run of consecutive even integers having exactly n divisors. %C A325116 The start of the first run of exactly k consecutive even integers having exactly n divisors is A325117(n,k). %C A325116 If m does not divide n, then a(n) < 2^(m-1). Proof: 2^(m-1) consecutive even integers must include one congruent to 2^(m-1) mod 2^m, and the number of divisors of this one is a multiple of m. %C A325116 For m > 2, if 2*(m-1) does not divide n, then a(n) < 2^(m-1). Proof: 2^(m-1) consecutive even integers must include two congruent to 2^(m-2) mod 2^(m-1). These are 2^(m-2)*x and 2^(m-2)*(x+2) for some odd x, and x and x+2 each have n/(m-1) divisors. If 2*(m-1) does not divide n, then n/(m-1) is odd. Any number with an odd number of divisors is a square, so x and x+2 are squares, but squares cannot differ by 2. %C A325116 14 <= a(24) <= 15. Dickson's conjecture implies a(24)=15. %Y A325116 Cf. A119479 (with consecutive integers), A319045 (with consecutive odd integers). %Y A325116 Cf. A325117. %K A325116 nonn,more,hard %O A325116 1,4 %A A325116 _David Wasserman_, Mar 27 2019