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).

12, 38, 80, 212, 224, 440, 440, 854, 1250, 1460, 1742, 2282, 2282, 3434, 4190, 4664, 4760, 4760, 6890, 8054, 8054, 8054, 12374, 12830, 12830, 13592, 13592, 14282, 17402, 17402, 18212, 22502, 22502, 22502, 25220, 28202, 28202, 32234, 32402, 32402, 38012

Offset

2,1

Comments

The sequence is nondecreasing. See comment in A242758.

a(n) >= prime(n)^2+3. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014

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

Links

Peter J. C. Moses, Table of n, a(n) for n = 2..2501

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014, (Section 10).

Formula

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246821). - Vladimir Shevelev, Sep 02 2014

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

Mathematica

lpf[n_] := FactorInteger[n][[1, 1]];
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]]];
Table[a[n], {n, 2, 50}] (* Jean-François Alcover, Nov 02 2018 *)

Prog

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

Crossrefs

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

Sequence in context: A039294 A043897 A079539 * A212510 A213490 A259517

Adjacent sequences: A242717 A242718 A242719 * A242721 A242722 A242723

Keyword

nonn

Author

Vladimir Shevelev, May 21 2014

Status

approved