A280877 Occurrences of decrease of the probability density P(a(n)) of coprime numbers k,m, satisfying 1 <= k <= a(n) and 1 <= m <= a(n); i.e., P(a(n)) < P(a(n)-1).

2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128

Offset

1,1

Comments

Probability densities satisfying P(a(n)) < P(a(n)-1).

A285022 is a subset.

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

The sequence is very regular in the sense that all {0 < i} 2i appear in this sequence, as well as all {0 < i} 30i - 15 appear in this sequence.

Presuming P(n) > 0.6: phi(n)/n < 1/2 for n congruent to 0 mod 2, P(n) < P(n-1).

Presuming P(n) > 0.6: phi(n)/n < 8/15 for n congruent to 15 mod 30, P(n) < P(n-1).

A280877 = {i > 0 | 2i} union {i > 0 | 30i - 15} union A280878 union A280879.

The irregular appearances are given in the two disjoint sequences A280878 and A280879.

See also A285022.

Experimental observation: n/a(n) < Euler constant (A001620).

Probability density P(a(n)) = A018805(a(n))/a(n)^2.

There seems, with good reason, to be a high correlation between the odd numbers in this sequence and A079814. - Peter Munn, Apr 11 2021

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..5682

Mark Kac, Statistical independence in probability, analysis and number theory pp. 53-79.

Mathematica

P[n_] := P[n] = (2 Sum[CoprimeQ[i, j] // Boole, {i, n}, {j, i-1}] + 1)/n^2;
Select[Range[2, 200], P[#] < P[#-1]&] (* Jean-François Alcover, Nov 15 2019 *)

Prog

(Python)
from fractions import gcd
t = 1
to = 1
i = 1
x = 1
while x < 10000:
....x = x + 1
....y = 0
....while y < x:
........y = y + 1
........if gcd(x, y) == 1:
............t = t + 2
....e = t*(x-1)*(x-1) - to*x*x
....if e < 0:
........print(i, x)
........i = i + 1
....to = t
(PARI) P(n) = sum(i=1, n, sum(j=1, n, gcd(i, j)==1))/n^2;
isok(n) = P(n) < P(n-1); \\ Michel Marcus, Jan 28 2017

Crossrefs

Cf. A001620, A018805, A079814, A279796, A280878, A285022.

Sequence in context: A319808 A055956 A161207 * A289509 A336735 A304711

Adjacent sequences: A280874 A280875 A280876 * A280878 A280879 A280880

Keyword

nonn

Author

A.H.M. Smeets, Jan 09 2017

Status

approved