A330899 -id:A330899 - 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

A385562 Numbers m such that (1/m) * Sum_{k=1..m} k/phi(k) sets a record value, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 22, 24, 30, 42, 60, 66, 72, 78, 84, 90, 114, 120, 150, 156, 180, 198, 210, 300, 330, 390, 420, 510, 546, 570, 600, 630, 750, 780, 840, 966, 990, 1122, 1170, 1200, 1260, 1410, 1470, 1560, 1596, 1620, 1650, 1680, 1806, 1830, 1890, 1980, 2100

Offset: 1

Amiram Eldar, Jul 03 2025

nonn

Limit_{m->oo} (1/m) * Sum_{k=1..m} k/phi(k) = zeta(2)*zeta(3)/zeta(6) (A082695) (Sitaramachandrarao, 1985; Sándor et al., 2005). This sequence is infinite if this mean converges to the limit only from below.

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 29.

Amiram Eldar, Table of n, a(n) for n = 1..1752 (terms below 10^10)
R. Sitaramachandrarao, On an error term of Landau. II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.

Cf. A000010, A068885, A069947, A082695, A330899, A385561.

seq[lim_] := Module[{s = {}, sum = 0, rm = 0, r}, Do[sum += k/EulerPhi[k]; r = sum/k; If[r > rm, rm = r; AppendTo[s, k]], {k, 1, lim}]; s]; seq[2500]

list(lim) = {my(sm = 0, rm = 0, r); for(k = 1, lim, sm += k/eulerphi(k); r = sm/k; if(r > rm, rm = r; print1(k, ", ")));}

A342100 Abundant numbers k at which the ratio (number of abundant numbers in 1..k)/k reaches a new record high.

Original entry on oeis.org

12, 18, 20, 24, 40, 42, 56, 60, 72, 80, 84, 88, 90, 102, 104, 108, 112, 114, 354, 366, 368, 372, 380, 384, 392, 396, 400, 402, 464, 468, 476, 480, 492, 500, 504, 552, 560, 564, 572, 576, 580, 582, 650, 654, 836, 840, 945, 948, 952, 954, 1002, 2002, 2004, 2024

Offset: 1

Jon E. Schoenfield, Feb 27 2021

Let rho(k) = (number of abundant numbers in 1..k)/k. According to A302991 ("Decimal expansion of the asymptotic density of abundant numbers"), lim_{k->infinity} rho(k) = 0.247619...
a(115) = 7254; rho(7254) = 1810/7254 = 0.2495175075820...
Conjecture: a(115) is the final term of this sequence.
This sequence is finite since rho(2212) > A302991 and therefore there is a number N such that abs(rho(n) - A302991) < eps for all n > N and for an arbitrarily small eps > 0. Therefore, the number of values of n for which rho(n) > rho(2212) is finite. - Amiram Eldar, Dec 06 2024

			k=12 is the 1st abundant number, so at k=12, rho(k) increases from 0 to 1/12 = 0.08333..., a record high, so a(1)=12.
k=18 is the 2nd abundant number, so at k=18, rho(k) reaches 2/18 = 1/9 = 0.11111..., the next record high, so a(2)=18.
k=20 is the 3rd abundant number, so at k=20, rho(k) reaches 3/20 = 0.15, the next record high, so a(3)=20.
k=24 is the 4th abundant number, so at k=24, rho(k) reaches 4/24 = 1/6 = 0.16666..., the next record high, so a(4)=24.
k=30 is the 5th abundant number, so at k=30, rho(k) again reaches 5/30 = 1/6; this is not a new record high, so 30 is not a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..115

Cf. A005101 (abundant numbers), A302991, A330899.

s = {}; c = 0; rm = 0; Do[If[DivisorSigma[1, n] > 2*n, c++; If[(r = c/n) > rm, rm = r; AppendTo[s, n]]], {n, 1, 10^4}]; s (* Amiram Eldar, Feb 28 2021 *)

Keyword "fini" added by Amiram Eldar, Dec 06 2024

A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 16, 22, 28, 36, 66, 96, 100, 126, 156, 190, 330, 430, 540, 820, 876, 1086, 1422, 10596, 10836, 18096, 35796, 55786, 69336, 111100, 168666, 284650, 905950, 1482300, 1745590, 2405560, 2661310, 4023306, 5869956, 17454580, 25670646, 51305346, 79969618, 211025650, 622626790

Offset: 1

Amiram Eldar, Jul 03 2025

nonn

6/Pi^2 is the asymptotic mean of phi(k)/k, i.e., lim_{m->oo} (1/m) * Sum_{k=1..m} phi(k)/k = 6/Pi^2 (Walfisz, 1963; Sándor et al., 2005).

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 27.
Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.

Cf. A000010, A059956, A071708, A072155, A330899, A385562.

seq[lim_] := Module[{s = {}, sum = 0, dm = 1, d}, Do[sum += EulerPhi[k]/k; If[(d = Abs[sum/k - 6/Pi^2]) < dm, dm = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]

list(lim) = {my(sm = 0, dm = 1, d); for(k = 1, lim, sm += eulerphi(k)/k; d = abs(sm/k - 6/Pi^2); if(d < dm, dm = d; print1(k, ", ")));}

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

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A385562 Numbers m such that (1/m) * Sum_{k=1..m} k/phi(k) sets a record value, where phi is the Euler totient function (A000010).

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A342100 Abundant numbers k at which the ratio (number of abundant numbers in 1..k)/k reaches a new record high.

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A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).

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