A201828 - OEIS

%I #41 Dec 03 2014 00:53:07

%S 37,37,2,2,2,2,907,2,2833,907,2,8269,2749,2953,5413,7699,2137,27103,

%T 28513,74377,45673,56629,79147,33529,15259,96847,101599,57649,44983,

%U 300973,706309,715357,351847,38557,308809,720607,901447,2229889,867253,2370937,1276867

%N The smallest A(m) such that the interval (A(m)*n, A(m+1)*n) contains exactly one element of A, where A is the sequence of primes p for which p-2 is not prime.

%C This sequence is the "A-analog" of A195871.

%C This is a possible model sequence to understand the role of twin primes in sequences like A195871.  In particular, if after a large number N_tw, there are no twin primes, what primes will take their place in A195871?  Our observations and expectations are expressed in the following conjecture.

%C Conjecture: For n>=13, every a(n) is the lesser of a pair of cousin primes p and p+4, cf. A023200.  Note that it is only conjectured that there are infinitely many pairs of cousin primes.

%C The limit of a(n) as n goes to infinity is infinity.

%e Let n=2. We have the following intervals of the form (2*p,2*q), where p,q are consecutive primes in A025584:(4,6),(6,22),(22,34),(34,46),(46,58),(58,74),(74,82),..., containing 0,2,2,2,2,3,1,... primes from A025584. The interval (74,82) is the first to contain exactly one prime from A025584, so a(2)=74/2=37.

%t myPrime=Select[#,!PrimeQ[#-2]&]&[Prime[Range[500000]]];  binarySearch[lst_,find_]:=Module[{lo=1,up=Length[lst],v},(While[lo<=up,v=Floor[(lo+up)/2];If[lst[[v]]-find==0,Return[v]];If[lst[[v]]<find,lo=v+1,up=v-1]];0)]; myPrimeQ[n_]:=binarySearch[myPrime,n]; nextMyPrime[n_,offset_Integer:1]:=myPrime[[myPrimeQ[NextPrime[n,NestWhile[#1+1&,1,!myPrimeQ[NextPrime[n,#1]]>0&]]]+offset-1]];   z=1;(*example for "contains exactly ONE myPrime in the interval"*)Table[myPrime[[NestWhile[#1+1&,1,!((nextMyPrime[n myPrime[[#1]],z]<n myPrime[[#1+1]]&&nextMyPrime[n myPrime[[#1]],z+1]>n myPrime[[#1+1]]))&]]],{n,2,30}]

%o (PARI) npr(n) = {local(p); p=n+1; while(!isprime(p) || isprime(p-2), p=p+1); p}

%o cnt(a,b) = {local(r); r=0; for(p=a, b, if(isprime(p) && !isprime(p-2), r=r+1)); r}

%o a201828(n) = {local(a,b); a=2; b=3; while(cnt(a*n, b*n) != 1, a=b; b=npr(b)); a} \\ _Michael B. Porter_, Feb 18 2013

%Y Cf. A025584, A195871, A023200.

%K nonn

%O 2,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jan 09 2013