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Let $n$ be a positive integer and $σ(n)$ the sum-of-divisors function. Then the abundancy index of $n$ is defined as

$$ I(n) = \displaystyle \frac{\sigma(n)}{n}. $$

If $I(n)=2$, $n$ is perfect. Since so far no odd perfect numbers have been found, it is interesting to find odd integers $n$ that ''come close'' to $I(n)=2$. Note that here I define ''closeness'' in a very loose sense as record values of $\displaystyle \delta_n = e^{\left| \log \frac{2}{I(n)} \right|}$.

To my surprise, I have not been able to find any computational results on this subject on the internet, neither in terms of ''records'', nor any abundancy index tables for odd positive integers. Can someone refer me to some publication/internet website/text file/OEIS page that lists such a table of records?

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  • $\begingroup$ There a plenty of odd numbers $n$ with $I(n)>2$, you know (see A005231). So do you consider "records" from either side? $\endgroup$ Oct 25 '19 at 8:55
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    $\begingroup$ Records from the deficient side: A188597 - And records from the abundant side: A188263 $\endgroup$ Oct 25 '19 at 8:59
  • $\begingroup$ @JeppeStigNielsen Thank you for those links. Is there a list of associated abundancy values somewhere as well? $\endgroup$
    – Klangen
    Oct 25 '19 at 9:05
  • $\begingroup$ Note that the term $n$ is almost perfect means that $\sigma(n)=2n-1$, so that $n$ is deficient by $1$/has deficiency $1$. $\endgroup$ Nov 2 '19 at 16:00
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The measure you propose apears to give A171929 which is $$1, 3, 9, 15, 45, 105, 315, 1155, 7425, 8415, 8925, 31815, 32445, 351351, 442365,\ldots$$ Also see my comments to the question above.

The first 28 members of this sequence give $I(n)$ values of: 1, 4/3, 13/9, 8/5, 26/15, 64/35, 208/105, 768/385, 992/495, 1872/935, 5952/2975, 7072/3535, 21632/10815, 78080/39039, 294912/147455, 3066752/1533385, 462592/231295, 723840/361921, 2714112/1357055, 4552704/2276351, 10469888/5234943, 12638208/6319105, 35340032/17670015, 543756288/271878145, 574758912/287379455, 1296422400/648211201, 1814828288/907414145, 1854935040/927467519 These are fractions. This is quick to find in PARI/GP. I also used PARI/GP to find the terms for looking up in OEIS.

Converted to approximate decimal representations, these $I(n)$ become: 1.0000000000000000000000000000000000000, 1.3333333333333333333333333333333333333, 1.4444444444444444444444444444444444445, 1.6000000000000000000000000000000000000, 1.7333333333333333333333333333333333333, 1.8285714285714285714285714285714285714, 1.9809523809523809523809523809523809524, 1.9948051948051948051948051948051948052, 2.0040404040404040404040404040404040404, 2.0021390374331550802139037433155080214, 2.0006722689075630252100840336134453782, 2.0005657708628005657708628005657708628, 2.0001849283402681460933888118354137772, 2.0000512308204615896923589231281538974, 2.0000135634600386558611101692041639822, 1.9999882612651095452218457856311363421, 2.0000086469659958062214920339825763635, 1.9999944739321564650849218475855228075, 2.0000014737796183647678244433718603889, 2.0000008785991264088886116420534443063, 2.0000003820480948885212312722411686240, 1.9999996834994829172802161065530640811, 2.0000001131860952013906043656442849652, 1.9999999926437632565133177585862960776, 2.0000000069594397414387190622238461688, 1.9999999969145858681328155574405138982, 1.9999999977959347327564526779555546823, 2.0000000021564097491590969667154456975

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