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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 (list; graph; refs; listen; history; text; internal format)
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: A117691 A243564 A171627 * A143487 A031350 A031353

Adjacent sequences:  A239881 A239882 A239883 * A239885 A239886 A239887

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Mar 28 2014

STATUS

approved

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Last modified January 17 18:48 EST 2019. Contains 319251 sequences. (Running on oeis4.)