

A238756


Number of ordered ways to write n = k + m (k > 0 and m > 0) such that 2*k + 1, prime(prime(k))  prime(k) + 1 and prime(prime(m))  prime(m) + 1 are all prime.


3



0, 1, 2, 3, 3, 2, 3, 3, 3, 4, 2, 5, 4, 3, 6, 4, 4, 3, 3, 6, 5, 5, 4, 6, 6, 5, 6, 2, 7, 5, 5, 6, 4, 4, 4, 5, 5, 8, 2, 5, 4, 5, 8, 2, 5, 2, 7, 4, 8, 6, 4, 5, 3, 8, 4, 7, 5, 3, 7, 7, 5, 7, 5, 7, 9, 8, 7, 5, 9, 7, 10, 9, 7, 7, 6, 9, 10, 4, 5, 5
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 1.
We have verified this for n up to 10^7.
The conjecture suggests that there are infinitely many primes p with 2*pi(p) + 1 and prime(p)  p + 1 both prime.


LINKS

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


EXAMPLE

a(6) = 2 since 6 = 2 + 4 with 2*2 + 1 = 5, prime(prime(2))  prime(2) + 1 = prime(3)  3 + 1 = 3 and prime(prime(4))  prime(4) + 1 = prime(7)  7 + 1 = 17  6 = 11 all prime, and 6 = 3 + 3 with 2*3 + 1 = 7 and prime(prime(3))  prime(3) + 1 = prime(5)  5 + 1 = 11  4 = 7 both prime.


MATHEMATICA

p[k_]:=PrimeQ[Prime[Prime[k]]Prime[k]+1]
a[n_]:=Sum[If[PrimeQ[2k+1]&&p[k]&&p[nk], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000040, A218829, A234694, A234695, A235189, A236832, A237413, A238134.
Sequence in context: A276859 A007538 A242285 * A025076 A110006 A289831
Adjacent sequences: A238753 A238754 A238755 * A238757 A238758 A238759


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 05 2014


STATUS

approved



