

A237769


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


7



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(np)  1, 3*pi(np) + 1 and 3*pi(np) + 5 are all prime. Also, for each integer n > 8, there is a prime p < n such that 3*pi(np)  1, 3*pi(np) + 1 and 3*pi(np)  5 are all prime.
(iii) For any integer n > 6, there is a prime p < n such that phi(np)  1 and phi(np) + 1 are both prime, where phi(.) is Euler's totient function.


LINKS

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


EXAMPLE

a(9) = 1 since 2, pi(92)  1 = 3 and pi(92) + 1 = 5 are all prime.
a(34) = 1 since 19, pi(3419)  1 = pi(15)  1 = 5 and pi(3419) + 1 = pi(15) + 1 = 7 are all prime.
a(35) = 1 since 19, pi(3519)  1 = pi(16)  1 = 5 and pi(3519) + 1 = pi(16) + 1 = 7 are all prime.


MATHEMATICA

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


CROSSREFS

Cf. A000010, A000040, A000720, A001359, A006512, A014574, A022004, A022005, A237705, A237706, A237768.
Sequence in context: A136032 A135975 A140361 * A187182 A176208 A330623
Adjacent sequences: A237766 A237767 A237768 * A237770 A237771 A237772


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 13 2014


STATUS

approved



