cp's OEIS Frontend

A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

%I A242719 #69 Mar 13 2020 11:35:04
%S A242719 10,26,50,170,170,362,362,842,842,1370,1370,1850,1850,2210,3722,3722,
%T A242719 3722,4892,5042,7082,7922,7922,7922,10610,10610,10610,11450,13844,
%U A242719 16130,16130,17162,19322,19322,24614,24614,25592,29504,29930,29930,36020,36020
%N A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).
%C A242719 The sequence is connected with a sufficient condition for the existence of an infinity of twin primes. In contrast to A242489, this sequence is nondecreasing.
%C A242719 All even numbers of the form A062326(n)^2 + 1 are in the sequence. All a(n)-1 are semiprimes. - _Vladimir Shevelev_, May 24 2014
%C A242719 a(n) <= A242489(n); a(n) >= prime(n)^2+1. Conjecture: a(n) <= prime(n)^4. - _Vladimir Shevelev_, Jun 01 2014
%C A242719 Conjecture: all numbers a(n)-3 are primes. _Peter J. C. Moses_ verified this conjecture up to a(2001) (cf. with conjecture in A242720). - _Vladimir Shevelev_, Jun 09 2014
%H A242719 Jinyuan Wang, <a href="/A242719/b242719.txt">Table of n, a(n) for n = 2..5000</a> (terms 2..2001 from Peter J. C. Moses).
%F A242719 Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246819). - _Vladimir Shevelev_, Sep 02 2014
%F A242719 a(n) = prime(n)^2 + 1 for and only for numbers n>=2 which are in A137291. - _Vladimir Shevelev_, Sep 04 2014
%t A242719 lpf[k_] := FactorInteger[k][[1, 1]];
%t A242719 a[n_] := a[n] = For[k = If[n == 2, 10, a[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
%t A242719 Array[a, 50, 2] (* _Jean-François Alcover_, Nov 06 2018 *)
%o A242719 (PARI)
%o A242719 lpf(k) = factorint(k)[1,1];
%o A242719 vector(50, n, k=6; while(lpf(k-3)<=lpf(k-1) || lpf(k-1)<prime(n+1), k+=2); k) \\ _Colin Barker_, Jun 01 2014
%Y A242719 Cf. A001359, A006512, A062326, A137291, A242489, A242490, A242847, A243960, A245363, A246501, A246748, A246819, A247011.
%K A242719 nonn
%O A242719 2,1
%A A242719 _Vladimir Shevelev_, May 21 2014