A084105 - OEIS

A084105

Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.

%I #20 Aug 07 2021 05:13:37

%S 3,29,113,139,199,523,1151,1669,2971,6947,10007,16141,25471,40639,

%T 79699,102761,173359,265621,404851,838249,1349533,1562051,6371537,

%U 7230479,27980987,42082303,53231051,70396589,192983851,253878617,390932389,465828731,516540163,1692327137

%N Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.

%C Are there entries other than a(3) for which the smaller difference exceeds 2?

%H Martin Ehrenstein, <a href="/A084105/b084105.txt">Table of n, a(n) for n = 1..57</a>

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a084105.txt">Maximally asymmetric prime triples, FORTRAN program</a>

%e a(3) = 113 because the ratio (113-109)/(127-113) = 2/7 = 0.28571.. is smaller than the previous minimum produced by (31-29)/(29-23) = 1/3 = 0.33333...

%o (PARI) a084105(limit)={my(p1=2,p2=3,r=0);forprime(p3=5,limit,my(q=max((p2-p1)/(p3-p2),(p3-p2)/(p2-p1)));if(q>r,r=q;print1(p2,", "));p1=p2;p2=p3)};

%o a084105(600000000) \\ _Hugo Pfoertner_, Sep 04 2020

%Y Cf. A084106, A031132, A008996, A002386, A329158, A329159, A329160, A329161, A337489.

%K nonn

%O 1,1

%A _Hugo Pfoertner_, May 29 2003

%E More terms from _Don Reble_, May 29 2003

%E a(32)-a(34) from _Hugo Pfoertner_, Nov 06 2019