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A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639). 15
6, 8, 14, 14, 20, 20, 32, 32, 32, 44, 44, 44, 62, 62, 62, 62, 74, 74, 74, 104, 104, 104, 104, 104, 104, 110, 110, 140, 140, 140, 140, 140, 152, 152, 182, 182, 182, 182, 182, 182, 194, 194, 200, 200, 230, 230, 230, 230, 242, 242, 242, 272, 272, 272, 272, 272 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.

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.

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

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

LINKS

Peter J. C. Moses, Table of n, a(n) for n = 2..10001

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

MATHEMATICA

lpf[k_] := FactorInteger[k][[1, 1]];

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]]];

Table[a[n], {n, 2, 100}] (* Jean-Fran├žois Alcover, Nov 03 2018 *)

PROG

(PARI)

lpf(k) = factorint(k)[1, 1];

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

CROSSREFS

Cf. A001359, A006512, A242489, A242490, A242720.

Sequence in context: A209686 A283286 A283375 * A315879 A315880 A315881

Adjacent sequences:  A242755 A242756 A242757 * A242759 A242760 A242761

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, May 22 2014

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)