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A242720 Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639). 29

%I

%S 12,38,80,212,224,440,440,854,1250,1460,1742,2282,2282,3434,4190,4664,

%T 4760,4760,6890,8054,8054,8054,12374,12830,12830,13592,13592,14282,

%U 17402,17402,18212,22502,22502,22502,25220,28202,28202,32234,32402,32402,38012

%N Smallest even k such that the pair {k-3,k-1} is not a twin prime pair and lpf(k-1) > lpf(k-3) >= prime(n), where lpf = least prime factor (A020639).

%C The sequence is nondecreasing. See comment in A242758.

%C a(n) >= prime(n)^2+3. Conjecture: a(n) <= prime(n)^4. - _Vladimir Shevelev_, Jun 01 2014

%C Conjecture. There are only a finite number of composite numbers of the form a(n)-1. _Peter J. C. Moses_ found only two: a(16)-1 = 4189 = 59*71 and a(20)-1 = 6889 = 83^2 and no others up to a(2501). Most likely, there are no others. - _Vladimir Shevelev_, Jun 09 2014

%H Peter J. C. Moses, <a href="/A242720/b242720.txt">Table of n, a(n) for n = 2..2501</a>

%H V. Shevelev, <a href="http://arXiv.org/abs/0912.4006">Theorems on twin primes-dual case</a>, arXiv:0912.4006 [math.GM], 2009-2014, (Section 10).

%F Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246821). - _Vladimir Shevelev_, Sep 02 2014

%F For n>=3, a(n) >= (prime(n)+1)^2 + 2. Equality holds for terms of A246824. - _Vladimir Shevelev_, Sep 04 2014

%t lpf[n_] := FactorInteger[n][[1, 1]];

%t Clear[a]; a[n_] := a[n] = For[k = If[n <= 2, 2, a[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 Table[a[n], {n, 2, 50}] (* _Jean-Fran├žois Alcover_, Nov 02 2018 *)

%o (PARI)

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

%o vector(60, n, k=6; while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)<prime(n+1), k+=2); k) \\ _Colin Barker_, Jun 01 2014

%Y Cf. A020639, A001359, A006512, A070155, A242489, A242490, A242758, A242847, A246748, A246821, A246824.

%K nonn

%O 2,1

%A _Vladimir Shevelev_, May 21 2014

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Last modified September 28 13:24 EDT 2020. Contains 337393 sequences. (Running on oeis4.)