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A238386 a(n) = |{0 < k < n-1: p = prime(k) + pi(n-k) 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 (list; graph; refs; listen; history; text; internal format)
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(n-k)^2 is prime.

We have verified part (i) of the conjecture for n up to 10^7.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(7) = 1 since prime(1) + pi(7-1) = 2 + 3 = 5 and 5 + 2 = 7 are both prime.

a(30) = 1 since prime(16) + pi(30-16) = 53 + 6 = 59 and 59 + 2 are both prime.

a(108) = 1 since prime(15) + pi(108-15) = 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[n-k]], 1, 0], {k, 1, n-2}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000040, A000720, A001359, A006512.

Sequence in context: A143111 A301367 A179195 * A064662 A024944 A304871

Adjacent sequences:  A238383 A238384 A238385 * A238387 A238388 A238389

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 26 2014

STATUS

approved

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Last modified October 17 08:07 EDT 2019. Contains 328106 sequences. (Running on oeis4.)