A280878 -id:A280878 - cp's OEIS frontend

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

A.H.M. Smeets, Jan 09 2017

nonn

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

A.H.M. Smeets, Table of n, a(n) for n = 1..5682
Mark Kac, Statistical independence in probability, analysis and number theory, Carus Monograph 12, Math. Assoc. Amer., 1959, pp. 53-79.

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

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 *)

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

from math 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

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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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).

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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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Python

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

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