

A237817


Number of primes p < n such that r = {q <= np: q and q + 2 are both prime} and r + 2 are both prime.


5



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 8, 7, 6, 6, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 4
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OFFSET

1,14


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 12.
(ii) For any integer n > 2, there is a prime p < n such that r = {q <= np: q and q + 2 are both prime} is a square.
See also A237815 for a similar conjecture involving Sophie Germain primes.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(13) = 1 since {q <= 13  2: q and q + 2 are both prime} = {3, 5, 11} has cardinality 3, and {3, 3 + 2} is a twin prime pair.


MATHEMATICA

TQ[n_]:=PrimeQ[n]&&PrimeQ[n+2]
sum[n_]:=Sum[If[PrimeQ[Prime[k]+2], 1, 0], {k, 1, PrimePi[n]}]
a[n_]:=Sum[If[TQ[sum[nPrime[k]]], 1, 0], {k, 1, PrimePi[n1]}]
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000040, A000290, A001359, A006512, A237705, A237706, A237769, A237815.
Sequence in context: A341151 A006546 A031268 * A306949 A189025 A236347
Adjacent sequences: A237814 A237815 A237816 * A237818 A237819 A237820


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 13 2014


STATUS

approved



