

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



1, 5, 7, 9, 21, 22, 44, 45, 66, 70, 78, 112, 150, 156, 160, 264, 270, 280, 432, 600, 1080, 1680
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Numbers n such that A000005(n) = floor(A007955(n)^(1/A000005(n))).
Numbers n such that A000005(n) = A000196(n).
Numbers n such that number of divisors of n equals number of squares <= n.
It is assumed that sequence is finite.
Numbers n such that A000196(n)/A000005(n) = 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 upperbound on the number of divisors of n 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. 485492.


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, or equivalent formulation, 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: A024571 A186406 A068332 * A268410 A029650 A049307
Adjacent sequences: A276731 A276732 A276733 * A276735 A276736 A276737


KEYWORD

nonn,fini,full


AUTHOR

Ilya Gutkovskiy, Oct 03 2016


STATUS

approved



