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

1,2

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

Links

Table of n, a(n) for n=1..22.

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.

Example

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.

Mathematica

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

Crossrefs

Cf. A000005, A000196, A007955, A280235.

Sequence in context: A186406 A370138 A068332 * A268410 A029650 A049307

Adjacent sequences: A276731 A276732 A276733 * A276735 A276736 A276737

Keyword

nonn,fini,full

Author

Ilya Gutkovskiy, Oct 03 2016

Status

approved