A280879 Occurrences of decrease of the probability density P(n) of coprime numbers k,m, satisfying 1 <= k <= a(n) and 1 <= m <= a(n), and a(n) congruent to 1 (mod 2) and a(n) not congruent to 3 (mod 6).

5005, 6545, 7315, 7735, 8645, 8855, 10465, 11165, 11935, 14245, 25025, 32725, 35035, 36575, 38675, 43225, 44275, 45815, 51205, 52325, 54145, 55055, 55825, 59675, 60515, 61985, 65065, 71225, 71995, 73255, 78155, 80465, 83545, 85085, 95095, 97405, 99715

Offset: 1

Author

A.H.M. Smeets, Jan 09 2017

Keywords

• nonn

Comments

Probability densities satisfying P(a(n)) < P(a(n)-1) and a(n) congruent to 1 (mod 2) and a(n) not congruent to 3 (mod 6).

It appears that most numbers satisfy a(n) congruent to 35 (mod 70), but a(74) congruent to 15 (mod 70) and a(93) congruent to 55 (mod 70).

Subset of A280877.

P(n) = ((2*Sum_{m=1..a(n)} phi(m))-1)/a(n)^2 (Cf. Euler phi function A000010).

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..102 from A.H.M. Smeets)

Crossrefs

Cf. A018805, A280877, A280878.

Programs

PARI

P(n) = (2 *sum(j=1, n, eulerphi(j)) - 1)/n^2;
isok(n) = (n % 2) && ((n % 6) != 3) && (P(n) < P(n-1)); \\ Michel Marcus, Jan 29 2017

Python

from fraction import gcd
t = 1
to = 1
i = 1
x = 1
while x > 0:
    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 and x%2 == 1 and x%6 != 3):
        print(i,x)
        i = i + 1
    to = t

Python

from sympy import totient
A280879_list, n, t = [], 1, 1
while len(A280879_list) < 1000:
    n += 1
    h = totient(n)
    t2 = t+h
    if n % 2 and n % 6 != 3 and 2*(n*(h*n - 2*t2 + 1) + t2) <  1:
        A280879_list.append(n)
    t = t2 # Chai Wah Wu, Feb 11 2017