A210465 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).

7, 13, 193, 271, 157, 193, 1297, 1741, 1231, 1033, 3541, 1447, 727, 2341, 9337, 1747, 9007, 2287, 3307, 14401, 8887, 8161, 8461, 28753, 23623, 23893, 10861, 59233, 70111, 28927, 44257, 101113, 152947, 41941, 65167, 41263, 183301, 409573, 150517, 35803, 138883, 81547, 79693

Offset: 2

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 22 2013

Keywords

• nonn

Comments

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

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.

Thus, if the conjectures are true, then there exist infinitely many triples of primes of the form {p,p+4,p+6}.

Crossrefs

Cf. A195325, A207820, A210467, A210475, A210476.

Programs

Mathematica

bPrime=Select[Table[Prime[n],{n,1000000}],Mod[#,3]==1&];(*A002476*)
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]]0&]]]+offset-1]];
z=1;(*example for "contains exactly ONE b-
primes"*)Table[bPrime[[NestWhile[#1+1&,1,!((nextBPrime[n bPrime[[#1]],z]n bPrime[[#1+1]]))&]]],{n,2,20}]