login
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
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.
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 28 2014
STATUS
approved