|
%I #24 May 30 2014 08:51:14
%S 10,26,50,254,170,392,362,944,842,1892,1370,2420,1850,2210,3764,6314,
%T 3722,4892,5042,7082,8612,9380,7922,12320,11414,10610,11450,13844,
%U 18872,16130,17162,20414,19322,26672,24614,25592,29504,37910,29930,44930,36020,36482
%N Smallest even k such that lpf(k-1) = prime(n), while lpf(k-3) > prime(n), where lpf=least prime factor (A020639).
%C This sequence is connected with a sufficient condition for the infinitude of twin primes.
%C Almost all numbers of the form a(n)-3 are primes. For composite numbers of such a form, see A242716.
%C Primes p for which a(p) = p^2+1 form sequence A062326 for p >= 3. - _Vladimir Shevelev_, May 21 2014
%H Peter J. C. Moses, <a href="/A242489/b242489.txt">Table of n, a(n) for n = 2..2501</a>
%F a(n) >= prime(n)^2+1. - _Vladimir Shevelev_, May 21 2014
%e Let n=2, prime(2)=3. Then lpf(10-1)=3, but lpf(10-3)=7>3.
%e Since k=10 is the smallest such k, then a(2)=10.
%t lpf[n_]:=lpf[n]=First[Select[Divisors[n],PrimeQ[#]&]];
%t Table[test=Prime[n];NestWhile[#+2&,test^2+1,!((lpf[#-1]==test)&&(lpf[#-3]>test))&],{n,2,60}] (* _Peter J. C. Moses_, May 21 2014 *)
%o (PARI) a(n) = {k = 6; p = prime(n); while ((factor(k-1)[1, 1] != p) || (factor(k-3)[1, 1] <= p), k+= 2); k;} \\ _Michel Marcus_, May 16 2014
%Y Cf. A001359, A006512.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, May 16 2014
%E More terms from _Michel Marcus_, May 16 2014
|
