|
%I #21 Feb 13 2013 23:58:29
%S 7,13,193,271,157,193,1297,1741,1231,1033,3541,1447,727,2341,9337,
%T 1747,9007,2287,3307,14401,8887,8161,8461,28753,23623,23893,10861,
%U 59233,70111,28927,44257,101113,152947,41941,65167,41263,183301,409573,150517,35803,138883,81547,79693
%N Let p_(3,1)(m) be the m-th prime == 1(mod 3). Then a(n) is the smallest p_(3,1)(m) such that the interval(p_(3,1)(m)*n, p_(3,1)(m+1)*n) contains exactly one prime == 1(mod 3).
%C The limit of a(n) as n goes to infinity is infinity.
%C Conjectures: (1) If q is the nearest prime>a(n), then q-a(n)=4 or 6 and both of these cases occur infinitely many times. (2) If q-a(n)=4 then q is the lesser of twin primes.
%C Thus, if the conjectures are true, then there exist infinitely many triples of primes of the form {p,p+4,p+6}.
%t bPrime=Select[Table[Prime[n],{n,1000000}],Mod[#,3]==1&];(*A002476*)
%t 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)];
%t bPrimeQ[n_]:=binarySearch[bPrime,n];
%t nextBPrime[n_,offset_Integer:1]:=bPrime[[bPrimeQ[NextPrime[n,NestWhile[#1+1&,1,!bPrimeQ[NextPrime[n,#1]]>0&]]]+offset-1]];
%t z=1;(*example for "contains exactly ONE b-
%t primes"*)Table[bPrime[[NestWhile[#1+1&,1,!((nextBPrime[n bPrime[[#1]],z]<n bPrime[[#1+1]]&&nextBPrime[n bPrime[[#1]],z+1]>n bPrime[[#1+1]]))&]]],{n,2,20}]
%Y Cf. A195325, A207820, A210467, A210475, A210476.
%K nonn
%O 2,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jan 22 2013
|
