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 A270592 #33 Feb 19 2019 18:21:17
%S A270592 2,3,3,5,23,29,18869,18899,1588761719,1588761929
%N A270592 List of pairs: smallest pair of successive primes p & q such that p+1 & q+1 have n distinct prime divisors in common.
%C A270592 All first 5 pairs are found and are proved to be a part of the sequence (using a computer program to find and prove them).
%C A270592 No other pairs were found yet.
%C A270592 The lower bound of the next pair (the 6th pair) is: 1425172824437699411. This is the lower end of the largest increasing primes gap confirmed (look up sequence A002386), that is because of that the gap is 1476 (look up sequence A005250), and the lowest difference possible for the 6th pair is 2310 or prime(5)# (look up sequence A002110), therefore the 2310 gap must be between larger primes than the pair of the maximal increasing prime gap, 1476.
%C A270592 The difference between the pairs is conjectured to be the primorials sequence: A002110.
%H A270592 Alon Kellner, <a href="/A270592/a270592.txt">Python program to calculate the sequence</a>
%H A270592 Alon Kellner, <a href="/A270592/a270592_1.txt">Python program to calculate the sequence (based on a conjecture, more efficient)</a>
%F A270592 p - The lower end of the sequence
%F A270592 q - The upper end of the sequence
%F A270592 prime - The primes sequence
%F A270592 x# - The primorial function for x
%F A270592 x|y - x is a divisor of y
%F A270592 p(n)=prime(m)
%F A270592 q(n)=prime(m+1)
%F A270592 q(n)-p(n)=prime(n)# (conjectured)
%F A270592 prime(n)#|(p(n)+1) (conjectured)
%F A270592 prime(n)#|(q(n)+1) (conjectured)
%e A270592 For n=2 the solution pair is p(2)=23, q(2)=29. p(2) & q(2) are successive primes, their consecutive numbers are p(2)+1=24, q(2)+1=30. Both 24 and 30 are multiples of both 2 and 3, which are two distinct primes. Since there are no smaller successive primes pair which is answering these criteria the pair (23, 29) is a part of the sequence, specifically the third pair in the sequence (n starts from 0).
%o A270592 (PARI) a(n) = {p = 2; f = factor(p+1)[,1]~; forprime(q=3, , g = factor(q+1)[,1]~; if (#setintersect(f, g) == n, return (p)); p = q; f = g;);}
%Y A270592 Cf. A000040, A002110, A002386, A005250, A270992.
%K A270592 nonn,tabf,more
%O A270592 0,1
%A A270592 _Alon Kellner_, Mar 19 2016