This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A323383 #30 Apr 16 2025 09:03:41 %S A323383 1,2,3,4,6,8,12 %N A323383 Proper divisors of 24. %C A323383 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. %e A323383 tau(1) = 1 >= 0.5 %e A323383 tau(2) = 2 >= 1 %e A323383 tau(3) = 2 >= 1.5 %e A323383 tau(4) = 3 >= 2 %e A323383 so 1, 2, 3, 4 are in the sequence. %e A323383 tau(5) = 2 < 2.5 %e A323383 so 5 is not in the sequence. %t A323383 Select[Range[10^3], 2 DivisorSigma[0, #] >= # &] (* _Michael De Vlieger_, Jan 20 2019 *) %o A323383 (PARI) for (n = 1, 100, if (sigma(n, 0) >= n / 2, print1(n, ", "))); %Y A323383 Cf. A000005, A368523. %Y A323383 Equals A018253 without 24. %K A323383 nonn,full,fini %O A323383 1,2 %A A323383 _Keith J. Bauer_, Jan 12 2019 %E A323383 Renamed by _Andrey Zabolotskiy_, Apr 15 2025