A237612 Least positive integer k such that A000720(k*n) is a square, or 0 if such a number k does not exist.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 10 2014

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

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

Programs

Mathematica

sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]
Do[Do[If[sq[k*n],Print[n," ",k];Goto[aa]],{k,1,Prime[n]-1}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]