cp's OEIS Frontend

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).

%I A280879 #31 May 22 2025 10:21:45
%S A280879 5005,6545,7315,7735,8645,8855,10465,11165,11935,14245,25025,32725,
%T A280879 35035,36575,38675,43225,44275,45815,51205,52325,54145,55055,55825,
%U A280879 59675,60515,61985,65065,71225,71995,73255,78155,80465,83545,85085,95095,97405,99715
%N 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).
%C A280879 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).
%C A280879 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).
%C A280879 Subset of A280877.
%C A280879 P(n) = ((2*Sum_{m=1..a(n)} phi(m))-1)/a(n)^2 (Cf. Euler phi function A000010).
%H A280879 Chai Wah Wu, <a href="/A280879/b280879.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..102 from A.H.M. Smeets)
%o A280879 (Python)
%o A280879 from fraction import gcd
%o A280879 t = 1
%o A280879 to = 1
%o A280879 i = 1
%o A280879 x = 1
%o A280879 while x > 0:
%o A280879     x = x + 1
%o A280879     y = 0
%o A280879     while y < x:
%o A280879         y = y + 1
%o A280879         if gcd(x,y) == 1:
%o A280879             t = t + 2
%o A280879     e = t*(x-1)*(x-1) - to*x*x
%o A280879     if (e < 0 and x%2 == 1 and x%6 != 3):
%o A280879         print(i,x)
%o A280879         i = i + 1
%o A280879     to = t
%o A280879 (PARI) P(n) = (2 *sum(j=1, n, eulerphi(j)) - 1)/n^2;
%o A280879 isok(n) = (n % 2) && ((n % 6) != 3) && (P(n) < P(n-1)); \\ _Michel Marcus_, Jan 29 2017
%o A280879 (Python)
%o A280879 from sympy import totient
%o A280879 A280879_list, n, t = [], 1, 1
%o A280879 while len(A280879_list) < 1000:
%o A280879     n += 1
%o A280879     h = totient(n)
%o A280879     t2 = t+h
%o A280879     if n % 2 and n % 6 != 3 and 2*(n*(h*n - 2*t2 + 1) + t2) <  1:
%o A280879         A280879_list.append(n)
%o A280879     t = t2 # _Chai Wah Wu_, Feb 11 2017
%Y A280879 Cf. A018805, A280877, A280878.
%K A280879 nonn
%O A280879 1,1
%A A280879 _A.H.M. Smeets_, Jan 09 2017