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).

1, 1, 3, 25, 67, 131, 1556, -1671

Offset: 3

Author

Vladimir Shevelev, Jun 13 2014

Keywords

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Comments

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.

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).

Links

• Vladimir Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014 (Sections 10,14).

Crossrefs

Cf. A020639, A243867, A242719, A242720, A242758, A243803, A243804.

Programs

PARI

lpf(k) = factorint(k)[1, 1];
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=p, u++)); r=s; s=t); u; } \\ Jinyuan Wang, Mar 13 2020

Extensions

More terms from Peter J. C. Moses, Jun 13 2014