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

1,5

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000290, A000720, A238902.

Sequence in context: A243564 A358329 A171627 * A143487 A031350 A031353

Adjacent sequences: A239881 A239882 A239883 * A239885 A239886 A239887

Keyword

nonn

Author

Zhi-Wei Sun, Mar 28 2014

Status

approved