cp's OEIS Frontend

A238902 a(n) = |{0 < k <= n: pi(pi(k*n)) is a square}|, where pi(x) denotes the number of primes not exceeding x.

%I A238902 #54 Mar 28 2014 22:40:43
%S A238902 1,2,1,1,2,3,2,1,2,4,3,4,3,3,3,2,5,5,4,3,5,4,5,4,5,5,6,4,4,6,4,5,4,6,
%T A238902 4,4,3,4,4,3,4,4,4,4,5,3,4,5,4,3,4,5,5,4,2,2,3,2,3,3,3,1,4,3,4,3,3,3,
%U A238902 5,2,1,2,3,5,3,4,4,2,1,5
%N A238902 a(n) = |{0 < k <= n: pi(pi(k*n)) is a square}|, where pi(x) denotes the number of primes not exceeding x.
%C A238902 Conjecture: (i) a(n) > 0 for all n > 0.
%C A238902 (ii) For every n = 1, 2, 3, ..., there exists a positive integer k <= (n+1)/2 such that pi(pi(k*n)) is a triangular number.
%C A238902 We have verified parts (i) and (ii) for n up to 2*10^5 and 10^5 respectively.
%C A238902 See A239884 for a sequence related to part (i) of the conjecture.
%H A238902 Zhi-Wei Sun, <a href="/A238902/b238902.txt">Table of n, a(n) for n = 1..10000</a>
%H A238902 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A238902 a(8)    = 1 since pi(pi(3*8)) = pi(pi(24)) = pi(9) = 2^2.
%e A238902 a(434)  = 1 since pi(pi(297*434)) = pi(pi(128898)) = pi(12064) = 38^2.
%e A238902 a(1042) = 1 since pi(pi(698*1042)) = pi(pi(727316)) = pi(58590) = 77^2.
%e A238902 a(9143) = 1 since pi(pi(8514*9143)) = pi(pi(77843502)) = pi(4550901) = 565^2.
%e A238902 a(48044)  > 0 since pi(pi(18332*48044))  = pi(45075237)  = 1650^2.
%e A238902 a(52158)  > 0 since pi(pi(27976*52158))  = pi(72792062)  = 2067^2.
%e A238902 a(78563)  > 0 since pi(pi(26031*78563))  = pi(100326489) = 2404^2.
%e A238902 a(98213)  > 0 since pi(pi(37308*98213))  = pi(174740922) = 3123^2.
%e A238902  a(141589) > 0 since pi(pi(42375*141589)) = pi(279538049)= 3899^2.
%e A238902 a(154473) > 0 since pi(pi(42954*154473)) = pi(307695484) = 4080^2.
%e A238902 a(195387) > 0 since pi(pi(60161*195387)) = pi(530982180) = 5282^2.
%t A238902 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A238902 p[k_,n_]:=SQ[PrimePi[PrimePi[k*n]]]
%t A238902 a[n_]:=Sum[If[p[k,n],1,0],{k,1,n}]
%t A238902 Table[a[n],{n,1,80}]
%o A238902 (PARI) {a(n) = sum( k=1, n, issquare( primepi( primepi( k*n))))}; /* _Michael Somos_, Mar 10 2014 */
%Y A238902 Cf. A000040, A000217, A000290, A000720, A237598, A237840, A238504, A239884.
%K A238902 nonn
%O A238902 1,2
%A A238902 _Zhi-Wei Sun_, Mar 06 2014