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 A323383 Positive integers k such that tau(k) >= k/2. 0
 1, 2, 3, 4, 6, 8, 12 (list; graph; refs; listen; history; text; internal format)
 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 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. LINKS 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

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Last modified December 11 21:21 EST 2019. Contains 329937 sequences. (Running on oeis4.)