A243593 Primes giving record values of f(n) = (2*Sum_{i=1..n}(i*prime(i)) / Sum_{i=1..n}(prime(i))-(n+1))/(n-1).

5, 7, 11, 13, 17, 23, 29, 37, 41, 53, 59, 97, 101, 127, 131, 137, 149, 223, 227, 307, 331, 337, 347, 349, 419, 541, 547, 557, 563, 569, 587, 809, 821, 967, 1277, 1361, 1367, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1847, 1861, 1867, 1871, 1949, 1973

Offset: 1

Author

Esko Ranta, Jun 07 2014

Keywords

• nonn

Comments

Is the sequence finite? It would mean that the value of f(n) would become monotonic after inclusion of the largest prime in the sequence.

It should be easy to prove that the value of lim 3*f(n) is 1 when n approaches infinity.

The generalized formula 3*(2*sum_XY/sum_Y - (n+1))/(n-1) is a non-linear correlation coefficient between the X (1,2,3...) and the nonnegative Y values, with range from -3 to +3, and linear correlation still giving value 1 or -1.

What is the next term after 32057?

Examples

			3rd prime is 5, and f(3) > f(2) so 5 is included in the sequence.
Starting at n=2, the values of f(n) are: 1/5, 3/10, 1/3, 11/28, 81/205, 71/174, 31/77, 81/200, 485/1161, ...
		

Links

• Esko Ranta, Table of n, a(n) for n = 1..206

Crossrefs

Cf. A000040, A014285, A007504, A046933, A014689, A000101.

Programs

PARI

f(n) = (2*sum(i=1,n,i*prime(i))/sum(i=1, n, prime(i)) - (n+1))/(n-1);
lista(nn) = {last = f(2); for (i=3, nn, new = f(i); if (new > last, print1(prime(i), ", ");); new = last;);} \\ Michel Marcus, Jun 10 2014