A323383 Proper divisors of 24.

1, 2, 3, 4, 6, 8, 12

Offset: 1

Author

Keith J. Bauer, Jan 12 2019

Keywords

• nonn
• full
• fini

Comments

These are the only 7 positive integers k which meet the constraint tau(k) >= k/2, where tau(k) = A000005(k). 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 j-th 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.

Examples

			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.
		

Crossrefs

Cf. A000005, A368523.

Equals A018253 without 24.

Programs

Mathematica

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

PARI

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

Extensions

Renamed by Andrey Zabolotskiy, Apr 15 2025