A072355 -id:A072355 - cp's OEIS frontend

A353537 Numbers whose abundancy index is larger than Pi^2/6.

4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150

Offset: 1

Amiram Eldar, Apr 25 2022

nonn

The abundancy index of a number k is sigma(k)/k, where sigma is the sum of divisors function (A000203).
Pi^2/6 (A013661) is the asymptotic mean of the abundancy indices of the positive integers.
The least odd term is 45 and the least term that is coprime to 6 is 25025.
Davenport (1933) proved that sigma(k)/k possesses a continuous distribution function and that the asymptotic density of numbers with abundancy index that is larger than x exists for all x > 1 and is a continuous function of x. Therefore, this sequence has an asymptotic density.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 41, 436, 4258, 42928, 428557, 4286145, 42864566, 428585795, 4286368677, 42861854540, ... Apparently, the asymptotic density is 0.4286... which means that the distribution of the abundancy indices is skewed with a positive nonparametric skew.

			4 is a term since sigma(4)/4 = 7/4 = 1.75 > Pi^2/6 = 1.644...

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Nonparametric skew.

Cf. A000203, A005100, A013661, A017665, A017666, A072355, A302991, A330899.
A005101 is a subsequence.
Subsequences: A353538, A353539, A353540, A353541.

Select[Range[150], DivisorSigma[-1, #] > Pi^2/6 &]

isok(k) = sigma(k)/k > Pi^2/6; \\ Michel Marcus, Apr 25 2022

A074920 Integers k such that phi(k) = 6k/Pi^2 rounded to nearest integer.

Original entry on oeis.org

1, 2, 3, 4, 33, 39, 99, 3237, 9711, 2069301, 6207903, 45502509, 24091595067

Offset: 1

Joseph L. Pe, Oct 01 2002

The average value of phi(n) can be approximated for large n by 6n/Pi^2 (Tattersall, p. 162).

			phi(99) = 60 and 6*99/Pi^2 = 60.1848...., which rounds off to 60. Hence 99 is a term of the sequence.

James J. Tattersall, "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001.

Cf. A000010, A072355.

Cf. A059956 (6/Pi^2).

s = Pi^2; l = {}; Do[ If[Abs[EulerPhi[n] - 6 n/ s] <= 0.5, l = Append[l, n]], {n, 1, 10^5}]; l

a(10)-a(13) from Amiram Eldar, May 10 2019

Name edited by Michel Marcus, May 10 2019

A308045 Numbers k such that usigma(k) = round(zeta(2)/zeta(3)*k), where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 2, 3, 4, 35, 44, 111, 123, 1105, 1900, 2920, 12452, 17889, 34200, 65067, 716148, 14134055, 179040201, 221709100, 221743300, 221766100, 221788900, 1120968741, 1272582040, 1441454511, 7339101375

Offset: 1

Amiram Eldar, May 10 2019

The unitary version of A072355.
zeta(2)/zeta(3) is the asymptotic mean of the unitary abundancy index usigma(k)/k (A306633).
a(27) > 10^10.

			35 is in the sequence since usigma(35) = 48, and (zeta(2)/zeta(3)) * 35 = 47.895... has a round value of 48.

Cf. A002117, A013661, A034448, A072355, A074920, A306633.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); meanAb = Zeta[2]/Zeta[3]; Select[Range[10^6], usigma[#] == Round[meanAb*#] &]

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A353537 Numbers whose abundancy index is larger than Pi^2/6.

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A074920 Integers k such that phi(k) = 6k/Pi^2 rounded to nearest integer.

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A308045 Numbers k such that usigma(k) = round(zeta(2)/zeta(3)*k), where usigma(k) is the sum of unitary divisors of k (A034448).

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