

A323383


Positive integers k such that tau(k) >= k/2.


0




OFFSET

1,2


COMMENTS

There are only 7 positive integers which meet this constraint. These happen to be proper divisors of 24. Inductively, there can only be a finite number of integers which meet this constraint. 1 has a perfect tau(k) / k ratio at 1. Every time a jth power of a prime is multiplied by it, its ratio is multiplied by (j + 1)/p^j. Although 2 also achieves a perfect score, the scores must degrade after 2 because the above ratio is less than 1 otherwise.


LINKS

Table of n, a(n) for n=1..7.


EXAMPLE

tau(1) = 1 >= 0.5
tau(2) = 2 >= 1
tau(3) = 2 >= 1.5
tau(4) = 3 >= 2
so 1, 2, 3, 4 are in the sequence.
tau(5) = 2 < 2.5
so 5 is not in the sequence.


MATHEMATICA

Select[Range[10^3], 2 DivisorSigma[0, #] >= # &] (* Michael De Vlieger, Jan 20 2019 *)


PROG

(PARI) for (n = 1, 100, if (sigma(n, 0) >= n / 2, print1(n, ", ")));


CROSSREFS

Cf. A000005.
Cf. A018253 without 24.
Sequence in context: A271487 A211397 A173542 * A085113 A129121 A018556
Adjacent sequences: A323380 A323381 A323382 * A323384 A323385 A323386


KEYWORD

nonn,full,fini


AUTHOR

Keith J. Bauer, Jan 12 2019


STATUS

approved



