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 A225868 #24 Apr 05 2021 20:42:48
%S A225868 6,9,12,13,16,19,24,31,32,48,53,83,89,107,113,131,139,149,167,179,191,
%T A225868 199,227,233,251,263,409,431,449,467,479,503,587,599,631,659,683,719,
%U A225868 769,827,983,1019,1091,1259,1367,1409,1439,1487,1499,1511,1583,1619,1979
%N A225868 Numbers m for which max_{2 <= k <= (m-2)/2} Sum_{d>1, d|m+k, k|m+d} 1 = 3.
%C A225868 Terms >= 53 are primes p such that p+2 is either prime or semiprime or, relatively rarely, the cube of a prime. However, according to calculations by _Peter J. C. Moses_, up to 4.2*10^13 there are no numbers p in the sequence for which p+2 is cube of a prime. One can prove that if such a prime p exists, then it is necessary (but not sufficient) for all numbers of the quadruple {r, 2*r - 1, 4*r^2 - 6*r + 3, (2*r - 1)^3 - 2} to be primes, where r == 19 (mod 30) is defined by the equality (2r-1)^3 - 2 = p. The first 3 suitable values of r are 229, 3109, and 17449. But the corresponding p's are not in the sequence. We conjecture that all primes of the sequence are Chen primes, that is, all of them are in A109611.
%H A225868 Peter J. C. Moses, <a href="/A225868/b225868.txt">Table of n, a(n) for n = 1..300</a>
%t A225868 f[n_] := (m = 0; Do[s = Sum[ Boole[ Divisible[n+d, k]], {d, Divisors[n+k] // Rest}]; If[s > m, m = s], {k, 2, (n-2)/2}]; m); Reap[ For[n = 1, n <= 2000, n = If[n < 53, n+1, NextPrime[n]], If[f[n] == 3, Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Jul 09 2013, after _Vladimir Shevelev_ *)
%Y A225868 Cf. A225867, A188579, A109611.
%K A225868 nonn
%O A225868 1,1
%A A225868 _Vladimir Shevelev_, May 18 2013