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A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

%I A242758 #74 Nov 03 2018 12:14:31
%S A242758 6,8,14,14,20,20,32,32,32,44,44,44,62,62,62,62,74,74,74,104,104,104,
%T A242758 104,104,104,110,110,140,140,140,140,140,152,152,182,182,182,182,182,
%U A242758 182,194,194,200,200,230,230,230,230,242,242,242,272,272,272,272,272
%N A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).
%C A242758 This is a version of A242720 with the absolute minima of k in the definition. The sequence is nondecreasing. Hypothetically, every pair {a(n)-3, a(n)-1} is a pair of twin primes.
%C A242758 If there exist infinitely many n such that a(n) < A242719(n) < a(n)^2, then from the result in the Shevelev link, it follows that for such n the set of numbers {even k: lpf(k-1) > lpf(k-3) >= prime(n)} either attains the absolute minimum of a(n) only in the case when {a(n)-3, a(n)-1} are twin primes, or does not attain it at all. Therefore, if there is only a finite number of twin primes, we have a contradiction. Thus the above condition is sufficient for infinity of twin primes.
%C A242758 Note also that, if there is only a finite number of twin primes, then after the last pair of them, this sequence will coincide with A242720. Then, in order to avoid a contradiction (again according to the Shevelev link), we should accept that there exists a number N_0 such that, for every n >= N_0, the following inequality holds: max(A242719(n),A242720(n)) > (min(A242719(n),A242720(n)))^2. - _Vladimir Shevelev_, May 24 2014
%C A242758 It is easy to prove that min(A242719(n), A242720(n)) >= prime(n)^2+1, while we conjecture that max(A242719(n), A242720(n)) <= prime(n)^4. Thus this conjecture implies there are infinitely many twin primes. - _Vladimir Shevelev_, Jun 01 2014
%H A242758 Peter J. C. Moses, <a href="/A242758/b242758.txt">Table of n, a(n) for n = 2..10001</a>
%H A242758 V. Shevelev, <a href="https://arxiv.org/abs/0912.4006">Theorems on twin primes-dual case</a>, arXiv:0912.4006 [math.GM], 2009-2014.
%t A242758 lpf[k_] := FactorInteger[k][[1, 1]];
%t A242758 a[n_] := a[n] = For[k = If[n == 2, 2, a[n-1]], True, k = k+2, If[lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
%t A242758 Table[a[n], {n, 2, 100}] (* _Jean-François Alcover_, Nov 03 2018 *)
%o A242758 (PARI)
%o A242758 lpf(k) = factorint(k)[1,1];
%o A242758 vector(100, n, k=6; while(lpf(k-1)<=lpf(k-3) || lpf(k-3)<prime(n+1), k+=2); k) \\ _Colin Barker_, Jun 01 2014
%Y A242758 Cf. A001359, A006512, A242489, A242490, A242720.
%K A242758 nonn
%O A242758 2,1
%A A242758 _Vladimir Shevelev_, May 22 2014