cp's OEIS Frontend

A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

%I A238890 #13 Feb 25 2025 20:55:20
%S A238890 1,2,2,2,1,1,2,2,2,2,4,1,2,2,3,6,1,1,4,4,1,5,3,5,5,4,5,1,2,5,7,6,5,2,
%T A238890 2,4,4,4,10,6,5,5,4,6,8,7,5,8,5,8,5,3,5,9,6,7,2,2,4,6,7,8,11,8,8,10,6,
%U A238890 8,10,2,5,11,7,5,10,10,8,7,9,8
%N A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.
%C A238890 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.
%C A238890 (ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.
%C A238890 The conjecture implies that there are infinitely many primes p with p - pi(pi(p)) (or p + pi(pi(p))) prime.
%H A238890 Zhi-Wei Sun, <a href="/A238890/b238890.txt">Table of n, a(n) for n = 1..3000</a>
%H A238890 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A238890 a(5) = 1 since prime(3*5) - pi(3*5) = 47 - 6 = 41 is prime.
%e A238890 a(28) = 1 since prime(18*28) - pi(18*28) = prime(504) - pi(504) = 3607 - 96 = 3511 is prime.
%t A238890 p[k_]:=PrimeQ[Prime[k]-PrimePi[k]]
%t A238890 a[n_]:=Sum[If[p[k*n],1,0],{k,1,n}]
%t A238890 Table[a[n],{n,1,80}]
%Y A238890 Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.
%K A238890 nonn
%O A238890 1,2
%A A238890 _Zhi-Wei Sun_, Mar 06 2014