A276734 Numbers n such that the number of divisors of n equals the integer part of the geometric mean of the divisors of n.

1, 5, 7, 9, 21, 22, 44, 45, 66, 70, 78, 112, 150, 156, 160, 264, 270, 280, 432, 600, 1080, 1680

Offset: 1

Author

Ilya Gutkovskiy, Oct 03 2016

Keywords

• nonn
• fini
• full

Comments

Numbers k such that A000005(k) = floor(A007955(k)^(1/A000005(k))).

Numbers k such that A000005(k) = A000196(k).

Numbers k such that the number of divisors of k equals the number of squares <= k.

It is assumed that the sequence is finite.

Numbers k such that A000196(k)/A000005(k) = r; r is a rational number. This sequence has r = 1. Does an r exist for which the sequence is infinite? - Ctibor O. Zizka, Jan 01 2017

The sequence is complete. This follows easily from the upper bound on the number of divisors of k proved by Nicolas & Robin. - Giovanni Resta, Jul 30 2018

Examples

			a(10) = 70, because 70 has 8 divisors {1, 2, 5, 7, 10, 14, 35, 70} and floor((1*2*5*7*10*14*35*70)^(1/8)) = floor(sqrt(70)) = 8; equivalently, we have 8 squares {1, 4, 9, 16, 25, 36, 49, 64} <= 70.
		

Links

• Ilya Gutkovskiy, Illustration of dynamics of floor(sqrt(n)) - sigma_0(n)
• L. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de N, Canadian Mathematical Bulletin 26 (1983), pp. 485-492.

Crossrefs

Cf. A000005, A000196, A007955, A280235.

Programs

Mathematica

Select[Range[10000], DivisorSigma[0, #1] == Floor[Sqrt[#1]] & ]