A237769 - OEIS

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A237769

Number of primes p < n with pi(n-p) - 1 and pi(n-p) + 1 both prime, where pi(.) is given by A000720.

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

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OFFSET

1,10

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 8, and a(n) = 1 only for n = 9, 34, 35.

(ii) For any integer n > 4, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(n-p) - 1, 3*pi(n-p) + 1 and 3*pi(n-p) - 5 are all prime.

(iii) For any integer n > 6, there is a prime p < n such that phi(n-p) - 1 and phi(n-p) + 1 are both prime, where phi(.) is Euler's totient function.

LINKS

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

EXAMPLE

a(9) = 1 since 2, pi(9-2) - 1 = 3 and pi(9-2) + 1 = 5 are all prime.

a(34) = 1 since 19, pi(34-19) - 1 = pi(15) - 1 = 5 and pi(34-19) + 1 = pi(15) + 1 = 7 are all prime.

a(35) = 1 since 19, pi(35-19) - 1 = pi(16) - 1 = 5 and pi(35-19) + 1 = pi(16) + 1 = 7 are all prime.

MATHEMATICA

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]

a[n_]:=Sum[If[TQ[PrimePi[n-Prime[k]]], 1, 0], {k, 1, PrimePi[n-1]}]

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

CROSSREFS