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 A242974 #72 Mar 21 2020 07:14:13
%S A242974 1,1,3,25,67,131,1556,-1671
%N A242974 Let M_n = A002110(n) (the n-th primorial), let N*(n)(N**(n), respectively) be the number of numbers k in [1, M_n] for which lpf(k-3) > lpf(k-1) >= prime(n) (lpf(k-1) > lpf(k-3) >= prime(n), respectively) such that k-3, k-1 are not twin primes, where lpf=least prime factor. Then a(n) = N*(n) - N**(n).
%C A242974 Small values of |a(n)| with respect to N*(n) + N**(n) (cf. A243867) clearly demonstrate the fact of statistical closeness of N*(n) and N**(n). See also comment in A243867.
%C A242974 If we don't exclude twin primes in the definition then, instead of this sequence, we would obtain -3, -14, -66, -443, -4569, -57422, -894506, -18465384, ... (cf. A000882). Thus twin primes strongly destroy the statistical closeness of N*(n) and N**(n).
%H A242974 Vladimir Shevelev, <a href="http://arXiv.org/abs/0912.4006">Theorems on twin primes-dual case</a>, arXiv:0912.4006 [math.GM], 2009-2014 (Sections 10,14).
%o A242974 (PARI) lpf(k) = factorint(k)[1, 1];
%o A242974 a(n) = {my(p=prime(n), r=1, s=2, t, u=0); for(k=4, prod(i=1, n, prime(i)), if((t=lpf(k-1))>r, if(r>=p&&(r<k-3||t<k-1), u--), if(t>=p, u++)); r=s; s=t); u; } \\ _Jinyuan Wang_, Mar 13 2020
%Y A242974 Cf. A020639, A243867, A242719, A242720, A242758, A243803, A243804.
%K A242974 sign,more
%O A242974 3,3
%A A242974 _Vladimir Shevelev_, Jun 13 2014
%E A242974 More terms from _Peter J. C. Moses_, Jun 13 2014