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 A282026 #47 May 23 2021 02:55:24
%S A282026 4,11,2,1,8,2,1,17,2,1,2,1,1,4,2,1,1,2,1,5,2,1,2,1,1,2,1,1,4,2,1,1,2,
%T A282026 1,4,2,1,1,2,1,2,1,1,2,1,1,1,2,1,8,2,1,8,2,1,2,1,1,1,1,1,1,2,1,2,1,1,
%U A282026 4,2,1,1,1,1,4,2,1,1,2,1,1,2,1,2,1,1,2,1
%N A282026 a(n) is the smallest m with gcd(m, 2*n+1) = 1 such that 2*n + 2*m + 1 is composite.
%C A282026 Starting at 2*n + 1, find the next odd composite number 2*n + 2*m + 1 that is relatively prime to 2*n + 1; then a(n) = m.
%C A282026 Since 2*n + 3 is relatively prime to 2*n + 1, and (2*n + 3)^2 is composite, a(n) <= 2*n^2 + 5*n + 4 (this is tight for n=0 and n=1).
%C A282026 From _Andrey Zabolotskiy_, Feb 13 2017: (Start)
%C A282026 Up to n = 10^7, a(n) are from the set [1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22]. First occurrence of 14 is a(99412), first occurrence of 22 is a(7225627). [Thanks to _Altug Alkan_ for pointing out a(99412).]
%C A282026 a(n) = 1 iff n is in A153238.
%C A282026 (End)
%C A282026 Based on _Altug Alkan_'s b-file, the records in this sequence are 4, 11, 17, 19, ... and occur at positions 1, 2, 8, 638, ... If the sequence is unbounded, then these two subsidiary sequences should be added to the OEIS (if they are new). - _N. J. A. Sloane_, Feb 13 2017
%H A282026 Altug Alkan, <a href="/A282026/b282026.txt">Table of n, a(n) for n = 0..10000</a>
%e A282026 When n=1, 2*n + 1 = 3, and 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 are all either prime or have a common factor with 3. The next term, 25, is OK, and so a(1) = (25 - 3)/2 = 11.
%t A282026 Table[m = 1; While[Nand[CoprimeQ[m, 2 n + 1], CompositeQ[2 (n + m) + 1]], m++]; m, {n, 0, 120}] (* _Michael De Vlieger_, Feb 18 2017 *)
%o A282026 (PARI) a(n) = my(k=1); while(isprime(2*n+2*k+1) || gcd(2*n+1, k) != 1, k++); k; \\ _Altug Alkan_, Feb 13 2017
%Y A282026 Cf. A153238, A282423, A282429.
%K A282026 nonn
%O A282026 0,1
%A A282026 _N. J. A. Sloane_, Feb 12 2017
%E A282026 Definition corrected by _Altug Alkan_, Feb 13 2017