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

2,1

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)=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}.

Links

Table of n, a(n) for n=2..44.

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]]<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, A207820, A210467, A210475, A210476.

Sequence in context: A178956 A319612 A247946 * A181492 A243368 A317209

Adjacent sequences: A210462 A210463 A210464 * A210466 A210467 A210468

Keyword

nonn

Author

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

Status

approved