

A238386


a(n) = {0 < k < n1: p = prime(k) + pi(nk) and p + 2 are both prime}, where pi(.) is given by A000720.


3



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

1,6


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 10.
(ii) For any integer n > 4, there is a positive integer k < n such that prime(k)^2 + pi(nk)^2 is prime.
We have verified part (i) of the conjecture for n up to 10^7.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(7) = 1 since prime(1) + pi(71) = 2 + 3 = 5 and 5 + 2 = 7 are both prime.
a(30) = 1 since prime(16) + pi(3016) = 53 + 6 = 59 and 59 + 2 are both prime.
a(108) = 1 since prime(15) + pi(10815) = 47 + 24 = 71 and 71 + 2 = 73 are both prime.


MATHEMATICA

tq[n_]:=PrimeQ[n]&&PrimeQ[n+2]
a[n_]:=Sum[If[tq[Prime[k]+PrimePi[nk]], 1, 0], {k, 1, n2}]
Table[a[n], {n, 1, 80}]


CROSSREFS

Cf. A000040, A000720, A001359, A006512.
Sequence in context: A301367 A334090 A179195 * A064662 A024944 A304871
Adjacent sequences: A238383 A238384 A238385 * A238387 A238388 A238389


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 26 2014


STATUS

approved



