

A226318


Positive integers n with p_{n+1}p_n = 2 and p_{n+3}p_{n+2} = 2, where p_k denotes the kth prime.


1



3, 5, 26, 33, 41, 43, 81, 140, 142, 171, 176, 234, 286, 294, 313, 318, 428, 458, 473, 475, 484, 577, 579, 584, 671, 743, 772, 862, 870, 872, 891, 934, 957, 1030, 1115, 1165, 1167, 1169, 1230, 1339, 1351, 1404, 1462, 1548, 1621, 1651, 1707, 1823, 1833, 1867, 1923, 2021, 2052, 2066, 2068, 2121, 2151, 2199, 2309, 2362
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OFFSET

1,1


COMMENTS

An old conjecture of de Polignac asserts that for any positive even integer d there are infinitely many n>0 with p_{n+1}p_n = d.
The author has formulated the following further extension.
Conjecture: For any positive even integers d_1,...,d_k, there are infinitely many positive integers n such that p_{n+2j1}p_{n+2j2} = d_j for all j=1,...,k.
For example,
p_{35209566+2j1}p_{35209566+2j2} = 2 for all j = 1,...,7,
p_{19726689+2j1}p_{19726689+2j2} = 6 for all j = 1,...,8,
and p_{297746+2j1}p_{297746+2j2} = 2j for j = 1,2,3,4,5.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, On functions taking only prime values, J. Number Theory 133(2013), 27942812.
ZhiWei Sun, On primes in arithmetic progressions, preprint, arXiv:1304.5988.
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Pages 11211174 from Volume 179 (2014), Issue 3.


EXAMPLE

a(1) = 3 and a(2) = 5 since {p_3,p_4}={5,7}, {p_5,p_6}={11,13} and {p_7,p_8}={17,19} are twin prime pairs.


MATHEMATICA

n=0
Do[If[Prime[k+1]Prime[k]==2&&Prime[k+3]Prime[k+2]==2, n=n+1;
Print[n, " ", k]], {k, 1, 100}]
PrimePi[#]&/@Transpose[Select[Partition[Prime[Range[2500]], 4, 1], #[[4]] #[[3]] == #[[2]]#[[1]]==2&]][[1]] (* Harvey P. Dale, Nov 20 2013 *)


PROG

(MAGMA) [n: n in [1..2500]  (NthPrime(n+1)NthPrime(n)) eq 2 and (NthPrime(n+3)NthPrime(n+2)) eq 2]; // Vincenzo Librandi, Jun 28 2015


CROSSREFS

Cf. A000040, A001359, A006512, A225889, A226115.
Sequence in context: A276968 A074701 A140127 * A154143 A101611 A268409
Adjacent sequences: A226315 A226316 A226317 * A226319 A226320 A226321


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jun 03 2013


STATUS

approved



