

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
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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. 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; 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: 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



