OFFSET
1,1
COMMENTS
The sequence is infinite, in view of a strong closeness between counting functions of numbers N_1 for which lpf(N_1-3) > lpf(N_1-1) >= prime(n) and numbers N_2 for which lpf(N_2-1) > lpf(N_2-3) >= prime(n), if {N_2-3, N_2-1} is not a pair of twin primes, where p_n=prime(n) and lpf=least prime factor (A020639). (Cf., for example, A243803-A243804). This closeness is explained by a somewhat symmetry (for details, see Shevelev's link).
However, it is very interesting to find an analytical proof of infinity of this and complementory sequences.
LINKS
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014 (Section 14).
MATHEMATICA
lpf[k_] := FactorInteger[k][[1, 1]];
a19[n_ /; n>1] := a19[n] = For[k = If[n == 2, 10, a19[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
a20[n_ /; n>1] := a20[n] = For[k = If[n <= 2, 2, a20[n-1]], True, k = k+2, If[Not[PrimeQ[k-3] && PrimeQ[k-1]] && lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Select[Range[250], a19[#] > a20[#]&] (* Jean-François Alcover, Nov 06 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Jun 02 2014
EXTENSIONS
More terms from Peter J. C. Moses, Jun 02 2014
STATUS
approved