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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).

%I A280877 #44 Jun 28 2025 12:55:28
%S A280877 2,4,6,8,10,12,14,15,16,18,20,21,22,24,26,28,30,32,33,34,36,38,40,42,
%T A280877 44,45,46,48,50,52,54,56,58,60,62,63,64,66,68,70,72,74,75,76,78,80,82,
%U A280877 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
%N 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).
%C A280877 Probability densities satisfying P(a(n)) < P(a(n)-1).
%C A280877 A285022 is a subset.
%C A280877 Related to Euler phi function by P(a(n)) = ((2*Sum_{1 <= m <= a(n)} phi(m))-1)/a(n)^2.
%C A280877 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.
%C A280877 Presuming P(n) > 0.6: phi(n)/n < 1/2 for n congruent to 0 mod 2, P(n) < P(n-1).
%C A280877 Presuming P(n) > 0.6: phi(n)/n < 8/15 for n congruent to 15 mod 30, P(n) < P(n-1).
%C A280877 A280877 = {i > 0 | 2i} union {i > 0 | 30i - 15} union A280878 union A280879.
%C A280877 The irregular appearances are given in the two disjoint sequences A280878 and A280879.
%C A280877 See also A285022.
%C A280877 Experimental observation: n/a(n) < Euler constant (A001620).
%C A280877 Probability density P(a(n)) = A018805(a(n))/a(n)^2.
%C A280877 There seems, with good reason, to be a high correlation between the odd numbers in this sequence and A079814. - _Peter Munn_, Apr 11 2021
%H A280877 A.H.M. Smeets, <a href="/A280877/b280877.txt">Table of n, a(n) for n = 1..5682</a>
%H A280877 Mark Kac, <a href="https://archive.org/details/statisticalindep0000kacm">Statistical independence in probability, analysis and number theory</a>, Carus Monograph 12, Math. Assoc. Amer., 1959, pp. 53-79.
%t A280877 P[n_] := P[n] = (2 Sum[CoprimeQ[i, j] // Boole, {i, n}, {j, i-1}] + 1)/n^2;
%t A280877 Select[Range[2, 200], P[#] < P[#-1]&] (* _Jean-François Alcover_, Nov 15 2019 *)
%o A280877 (Python)
%o A280877 from math import gcd
%o A280877 t = 1
%o A280877 to = 1
%o A280877 i = 1
%o A280877 x = 1
%o A280877 while x < 10000:
%o A280877     x = x + 1
%o A280877     y = 0
%o A280877     while y < x:
%o A280877         y = y + 1
%o A280877         if gcd(x,y) == 1:
%o A280877             t = t + 2
%o A280877     e = t*(x-1)*(x-1) - to*x*x
%o A280877     if e < 0:
%o A280877         print(i,x)
%o A280877         i = i + 1
%o A280877     to = t
%o A280877 (PARI) P(n) = sum(i=1, n, sum(j=1, n, gcd(i,j)==1))/n^2;
%o A280877 isok(n) = P(n) < P(n-1); \\ _Michel Marcus_, Jan 28 2017
%Y A280877 Cf. A001620, A018805, A079814, A279796, A280878, A285022.
%K A280877 nonn
%O A280877 1,1
%A A280877 _A.H.M. Smeets_, Jan 09 2017