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%I #37 Nov 06 2018 03:00:37
%S 19,35,38,41,45,50,53,56,57,58,59,63,76,77,78,79,80,81,83,84,85,92,93,
%T 95,96,108,109,112,113,116,117,124,125,126,142,143,146,154,157,173,
%U 184,185,186,193,194,195,196,197,203,215,217,224,227,232,233,237,241
%N Numbers n for which A242719(n) > A242720(n).
%C 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).
%C However, it is very interesting to find an analytical proof of infinity of this and complementory sequences.
%H V. Shevelev, <a href="https://arxiv.org/abs/0912.4006">Theorems on twin primes-dual case</a>, arXiv:0912.4006 [math.GM], 2009-2014 (Section 14).
%t lpf[k_] := FactorInteger[k][[1, 1]];
%t 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]]];
%t 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]]];
%t Select[Range[250], a19[#] > a20[#]&] (* _Jean-François Alcover_, Nov 06 2018 *)
%Y Cf. A242719, A242720, A242758.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Jun 02 2014
%E More terms from _Peter J. C. Moses_, Jun 02 2014
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