A353616 -id:A353616 - cp's OEIS frontend

A353617 Decimal expansion of the asymptotic median of the abundancy indices of the positive integers.

1, 5, 2, 3, 8, 1

Offset: 1

Amiram Eldar, Apr 30 2022

The abundancy index of a number k is sigma(k)/k = A017665(k)/A017666(k), where sigma is the sum-of-divisors function (A000203).
Davenport (1933) proved that sigma(k)/k possesses a continuous distribution function. Therefore, it has an asymptotic median.
The asymptotic mean of the abundancy indices is Pi^2/6 = 1.64493... (A013661).
Mitsuo Kobayashi (unpublished, 2018) found that the median is in the interval (1.523812, 1.5238175) (see the MathOverflow link).

			1.52381...

Harold Davenport, Über numeri abundantes, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, No. 6 (1933), pp. 830-837.

Sébastien Palcoux, On the density map of the abundancy index, MathOverflow, 2020.

Cf. A000203, A013661, A017665, A017666, A302991, A353615, A353616.

A353615 a(n) is the numerator of the median abundancy index of the first 2^n + 1 positive integers.

Original entry on oeis.org

4, 4, 13, 3, 32, 45, 63, 90, 111, 162, 210, 237, 7502, 11776, 240, 2944, 3008, 11648, 225152, 478080, 2704000, 1548160, 2462400, 10128256, 4999296, 8640768, 48264064, 32347648

Offset: 1

Amiram Eldar, Apr 30 2022

The abundancy index of a number k is sigma(k)/k = A017665(k)/A017666(k), where sigma is the sum-of-divisors function (A000203).

			The fractions begin with 4/3, 4/3, 13/9, 3/2, 32/21, 45/29, 63/41, 90/59, 111/73, 162/107, ...
The abundancy indices of the first 3 positive integers are 1, 3/2 and 4/3. The median is a(1)/A353616(1) = 4/3.

Cf. A000203, A017665, A017666, A353616 (denominators), A353617.

With[{m = 20}, t = Table[DivisorSigma[-1, n], {n, 1, 2^m + 1}]; Numerator @ Table[Median[t[[1 ;; 2^n + 1]]], {n, 1, m}]]

Limit_{n->oo} a(n)/A353616(n) = A353617.

cp's OEIS Frontend

A353617 Decimal expansion of the asymptotic median of the abundancy indices of the positive integers.

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