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820, 1276, 1422, 1926, 2080, 2640, 3160, 3186, 3250, 4446, 4720, 4930, 5370, 6006, 6546, 7386, 7450, 7476, 9066, 9276, 10626, 10836, 13146, 13300, 15640, 15666, 16056, 16060, 16446, 17020, 17466, 17550, 17766, 18040, 18910, 19176, 19230, 19416, 20736, 21000, 21246, 21310, 21376, 21726

820, 1276, 1422, 1926, 2080, 2640, 3160, 3186, 3250, 4446, 4720, 4930, 5370, 6006

Related to Euler phi function by ((2*Sum_{n=1..a(m)} phi(n))-1)/a(m)^2 <= 6/Pi^2.

It appears that a(n) mod 2 = 0, but this is not necessarily true. In fact we can only assure that if a(n) is odd, a(n)+1 will be in the sequence too. Proof: Presuming P(a(n)) > 0.5 for an odd a(n), by phi(a(n)+1) <= 0.5 for a(n)+1 being even, P(a(n)+1) will always be smaller than P(a(n)). So, odd appearances may occur in this sequence as well, where we only can state that if a(n) is odd, a(n)+1 will occur too. Moreover it seems that a(n) mod 10 = 0, 2, 6, but a(433) = 226528 is a counterexample to this conjecture.

a(1) = 820 is possibly the first occurrence. The average behavior of the probability density P(n) is given by 1/zeta(n,2), where zeta(n,2) is the finite expansion of Euler's zeta function, i.e., zeta(n,2) = Sum_{i=1..n} 1/i^2. So the average behavior is that P(n) tends to 1/zeta(2) = 6/Pi^2 from above. However, due to the irregular behavior of primality and composite numbers, P(n) behaves similarly, giving the first irregular occurrence satisfying P(n) < 6/Pi^2 at n = 820. On average, the occurrences have a density of something less than 0.0025, which would imply that we could expect the first occurrence near 400, but from the average behavior of the probability density P(n) it is about twice this value.

Python program#

import math

from fraction import gcd

pi = 2*math.acos(0)

error = 0.1

t = 1

x = 0

while x < 117500:

....x = x + 1

....y = 0

....while y < x:

........y = y + 1

........if gcd(x, y) == 1:

............t = t + 2

....error = t/x/x - 6/pi/pi

....if error < 0:

........print(x, t/x/x, error)

nonn,dead,changed

dead

A.H.M. Smeets, Dec 19 2016

proposed

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proposed

820, 1276, 1422, 1926, 2080, 2640, 3160, 3186, 3250, 4446, 4720, 4930, 5370, 6006, 6546, 7386, 7450, 7476, 9066, 9276, 10626, 10836, 13146, 13300, 15640, 15666, 16056, 16060, 16446, 17020, 17466, 17550, 17766, 18040, 18910, 19176, 19230, 19416, 20736, 21000, 21246, 21310, 21376, 21726

The corrected version of this sequence is a duplicate of A285022. - Jens Kruse Andersen, Jan 08 2021

Irregular occurrences of the probability density P(n) of coprime numbers k,m, satisfying 1 <= k <= a(n) and 1 <= m <= a(n), such that P(k) < 1/zeta(2) = 6/Pi^2.

Erroneous duplicate of A285022.

nonn,dead,changed

Program corrected and false term removed by Jens Kruse Andersen, Jan 08 2021

proposed

editing

editing

proposed

820, 1276, 1422, 1926, 2080, 2640, 3160, 3186, 3250, 4446, 4720, 4930, 5370, 6006, 6546, 7386, 7450, 7476, 9066, 9276, 10626, 10836, 13146, 13300, 15640, 15666, 16056, 16060, 16446, 17020, 17466, 17550, 17766, 18040, 18910, 19176, 19230, 19416, 20736, 21000, 21246, 21310, 21376, 21726

The corrected version of this sequence is a duplicate of A285022. - Jens Kruse Andersen, Jan 08 2021

A. H. M. Smeets, <a href="/A279796/b279796.txt">Table of n, a(n) for n = 1..228</a>

x = 10

Program corrected and false term removed by Jens Kruse Andersen, Jan 08 2021

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proposed

approved

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proposed