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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). 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 (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) = 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 A003110

Adjacent sequences:  A210464 A210465 A210466 * A210468 A210469 A210470

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified January 17 18:14 EST 2020. Contains 330987 sequences. (Running on oeis4.)