A237612 - OEIS

%I #12 Apr 06 2014 22:19:36

%S 1,1,3,2,2,4,1,1,1,1,5,2,2,2,28,34,9,3,3,5,20,7,1,1,1,1,1,1,2,14,5,17,

%T 3,16,12,23,18,4,4,30,46,10,50,23,36,18,40,14,2,2,3,3,1,1,1,1,1,1,32,

%U 7,11,68,19,79,29,267,10,8,12,6

%N Least positive integer k such that A000720(k*n) is a square, or 0 if such a number k does not exist.

%C According to the conjecture in A237598, we should always have 0 < a(n) < prime(n).

%H Zhi-Wei Sun, <a href="/A237612/b237612.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(3) = 3 since A000720(3*3) = 4 is a square, but neither A000720(1*3) = 2  nor A000720(2*3) = 3 is a square.

%t sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]

%t Do[Do[If[sq[k*n],Print[n," ",k];Goto[aa]],{k,1,Prime[n]-1}];

%t Print[n," ",0];Label[aa];Continue,{n,1,100}]

%Y Cf. A000290, A000720, A237578, A237597, A237598, A237614.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Feb 10 2014