A239884 Least positive integer k <= n with pi(pi(k*n)) a square, or 0 if such a number k does not exist, where pi(x) denotes the number of primes not exceeding x.

1, 1, 1, 1, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 7, 7, 7, 13, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 16, 9, 9, 9, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 13, 70, 20, 7, 7, 7, 7, 7, 7, 63, 11

Offset: 1

Author

Zhi-Wei Sun, Mar 28 2014

Keywords

• nonn

Comments

According to part (i) of the conjecture in A238902, a(n) should be always positive. We have verified this for all n = 1, ..., 2*10^5.

Examples

			a(5) = 4 since pi(pi(4*5)) = pi(8) = 2^2, but none of pi(pi(1*5)) = pi(3) = 2, pi(pi(2*5)) = pi(4) = 2 and pi(pi(3*5)) = pi(6) = 3 is a square.
a(192969) = 83187 with pi(pi(83187*192969)) = pi(715034817) = 6082^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, List of (n, a(n), sqrt(pi(pi(a(n)*n))) for n = 1, ..., 2*10^5
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Crossrefs

Cf. A000040, A000290, A000720, A238902.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[PrimePi[n]]
Do[Do[If[SQ[f[k*n]],Print[n," ",k];Goto[aa]],{k,1,n}];
Print[n," ",0];Label[aa];Continue,{n,1,80}]