A280877 - OEIS

%I #38 Apr 16 2021 00:00:55

%S 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 44,45,46,48,50,52,54,56,58,60,62,63,64,66,68,70,72,74,75,76,78,80,82,

%U 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 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 Probability densities satisfying P(a(n)) < P(a(n)-1).

%C A285022 is a subset.

%C Related to Euler phi function by P(a(n)) = ((2*Sum_{1 <= m <= a(n)} phi(m))-1)/a(n)^2.

%C 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 Presuming P(n) > 0.6: phi(n)/n < 1/2 for n congruent to 0 mod 2, P(n) < P(n-1).

%C Presuming P(n) > 0.6: phi(n)/n < 8/15 for n congruent to 15 mod 30, P(n) < P(n-1).

%C A280877 = {i > 0 | 2i} union {i > 0 | 30i - 15} union A280878 union A280879.

%C The irregular appearances are given in the two disjoint sequences A280878 and A280879.

%C See also A285022.

%C Experimental observation: n/a(n) < Euler constant (A001620).

%C Probability density P(a(n)) = A018805(a(n))/a(n)^2.

%C 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 A.H.M. Smeets, <a href="/A280877/b280877.txt">Table of n, a(n) for n = 1..5682</a>

%H Mark Kac, <a href="http://www.gibbs.if.usp.br/~marchett/estocastica/MarkKac-Statistical-Independence.pdf"> Statistical independence in probability, analysis and number theory</a> pp. 53-79.

%t P[n_] := P[n] = (2 Sum[CoprimeQ[i, j] // Boole, {i, n}, {j, i-1}] + 1)/n^2;

%t Select[Range[2, 200], P[#] < P[#-1]&] (* _Jean-François Alcover_, Nov 15 2019 *)

%o (Python)

%o from fractions import gcd

%o t = 1

%o to = 1

%o i = 1

%o x = 1

%o while x < 10000:

%o ....x = x + 1

%o ....y = 0

%o ....while y < x:

%o ........y = y + 1

%o ........if gcd(x,y) == 1:

%o ............t = t + 2

%o ....e = t*(x-1)*(x-1) - to*x*x

%o ....if e < 0:

%o ........print(i,x)

%o ........i = i + 1

%o ....to = t

%o (PARI) P(n) = sum(i=1, n, sum(j=1, n, gcd(i,j)==1))/n^2;

%o isok(n) = P(n) < P(n-1); \\ _Michel Marcus_, Jan 28 2017

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

%K nonn

%O 1,1

%A _A.H.M. Smeets_, Jan 09 2017