A280877 -id:A280877 - cp's OEIS frontend

A280878 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) congruent to {3,9,21,27}(mod 30).

21, 33, 63, 99, 147, 189, 231, 273, 297, 357, 363, 399, 429, 441, 483, 561, 567, 609, 627, 651, 663, 693, 741, 759, 777, 819, 861, 891, 897, 903, 957, 969, 987, 1023, 1029, 1071, 1089, 1113, 1131, 1173, 1197, 1209, 1221, 1239, 1281, 1287, 1311, 1323, 1353, 1407, 1419, 1443, 1449

Offset: 1

A.H.M. Smeets, Jan 09 2017

nonn

Probability densities satisfying P(a(n)) < P(a(n)-1) and 1 <= m <= a(n), and a(n) congruent to 1(mod 2) and a(n) congruent to {3,9,21,27}(mod 30).
Subset of A280877.
Related to Euler phi function A000010 by P(n) = ((2*Sum_{m=1..a(n)} phi(m))-1)/a(n)^2.

A.H.M. Smeets, Table of n, a(n) for n = 1..11142

from math import gcd
t = 1
to = 1
i = 1
x = 1
while x < 1450:
    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 and x%30 != 15):
        print(i,x)
        i = i + 1
    to = t

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).

Original entry on oeis.org

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

A.H.M. Smeets, Jan 09 2017

nonn

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).

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

Cf. A018805, A280877, A280878.

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

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

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

A079814 Odd integers k such that phi(k)/k < 6/Pi^2 where phi = A000010.

Original entry on oeis.org

15, 21, 33, 45, 63, 75, 99, 105, 135, 147, 165, 189, 195, 225, 231, 255, 273, 285, 297, 315, 345, 357, 363, 375, 399, 405, 429, 435, 441, 465, 483, 495, 525, 555, 561, 567, 585, 609, 615, 627, 645, 651, 663, 675, 693, 705, 735, 741, 759, 765, 777, 795, 819

Offset: 1

Matthew Vandermast, Feb 19 2003

Since, as Euler proved, the random chance of two integers not having a common prime factor is 6/Pi^2, these are the odd integers that share common factors with an above average fraction of integers. Is it known, or can it be calculated, what portion of odd integers satisfy this condition? (All even numbers qualify; for all multiples of 2, phi(n)/n <= 0.5.)

The sequence is closed under multiplication by any odd number. If we include the even numbers, the sequence of primitive terms begins 2, 15, 21, 33, 663, ... . - Peter Munn, Apr 11 2021

			phi(33)/33 = 20/33 or 0.6060606...; 6/Pi^2 is 0.6079271....

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010 (Euler totient function phi(n)), A280877, A280878, A280879.

Select[Range[1, 1000, 2], EulerPhi[#]/# < 6/Pi^2 &] (* Paolo Xausa, Aug 29 2025 *)

is(n)=n%2 && eulerphi(n)/n<6/Pi^2 \\ Charles R Greathouse IV, Sep 13 2013

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A280878 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) congruent to {3,9,21,27}(mod 30).

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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).

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A079814 Odd integers k such that phi(k)/k < 6/Pi^2 where phi = A000010.

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