A242847 Numbers n for which A242719(n) > A242720(n).

19, 35, 38, 41, 45, 50, 53, 56, 57, 58, 59, 63, 76, 77, 78, 79, 80, 81, 83, 84, 85, 92, 93, 95, 96, 108, 109, 112, 113, 116, 117, 124, 125, 126, 142, 143, 146, 154, 157, 173, 184, 185, 186, 193, 194, 195, 196, 197, 203, 215, 217, 224, 227, 232, 233, 237, 241

Offset: 1

Author

Vladimir Shevelev, Jun 02 2014

Keywords

• nonn

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).

Crossrefs

Cf. A242719, A242720, A242758.

Programs

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 *)

Extensions

More terms from Peter J. C. Moses, Jun 02 2014