A210467 Let p_(3,2)(m) be the m-th prime == 2(mod 3). Then a(n) is the smallest p_(3,2)(m) such that the interval(p_(3,2)(m)*n, p_(3,2)(m+1)*n) contains exactly one prime == 2 (mod 3).

2, 2, 101, 263, 1097, 251, 311, 461, 641, 941, 1601, 2351, 2543, 5003, 2837, 4787, 5711, 4283, 7901, 10331, 8831, 2687, 7877, 54287, 5711, 5501, 5303, 56087, 69827, 15641, 63611, 138581, 106427, 91571, 69827, 266177, 142421, 177533, 179687, 309311, 55691, 119291, 509543, 593987, 1393913

Offset

2,1

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) = 2 or 6 and both of these cases occur infinitely many times. (2) If q-a(n) = 2, then also q is lesser of a pair of cousin primes q and q+4, see A023200.

Thus, if the conjectures are true, then there exist infinitely many triples of primes of the form {p,p+2,p+6}.

Links

Table of n, a(n) for n=2..46.

Mathematica

bPrime=Select[Table[Prime[n], {n, 1000000}], Mod[#, 3]==2&];
binarySearch[lst_, find_]:=Module[{lo=2, 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)];
bPrimeQ[n_]:=binarySearch[bPrime, n];
nextBPrime[n_, offset_Integer:1]:=bPrime[[bPrimeQ[NextPrime[n, NestWhile[#1+1&, 1, !bPrimeQ[NextPrime[n, #1]]>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]]&&nextBPrime[n bPrime[[#1]], z+1]>n bPrime[[#1+1]]))&]]], {n, 2, 20}]

Crossrefs

Cf. A195325, A210465, A207820, A210475, A210476.

Sequence in context: A133295 A055470 A270591 * A156524 A194027 A376440

Adjacent sequences: A210464 A210465 A210466 * A210468 A210469 A210470

Keyword

nonn

Author

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

Status

approved