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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). 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 20 02:22 EDT 2019. Contains 323411 sequences. (Running on oeis4.)