

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.


2



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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, List of (n, a(n), sqrt(pi(pi(a(n)*n))) for n = 1, ..., 2*10^5
ZhiWei 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: A117691 A243564 A171627 * A143487 A031350 A031353
Adjacent sequences: A239881 A239882 A239883 * A239885 A239886 A239887


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 28 2014


STATUS

approved



