A301599 - OEIS

A301599

Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

%I #13 Mar 28 2018 05:14:21

%S 1,2,3,4,5,7,9,10,12,17,25,31,35,48

%N Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

%C Equivalently, define the function f(k) = k*prime(k)/Sum_{j=1..k} prime(j); sequence lists numbers k such that f(k) > f(m) for all m < k.

%C a(14)=48 is the final term. Beyond k=48, r(k) decreases fairly smoothly (although nonmonotonically); see the Example section.

%C For m = 4..18, the first k > 48 at which r(k) < 2 - 1/m is 50, 53, 61, 775, 2678, 8973, 23483, 63535, 159863, 431988, 1091840, 2753459, 7186422, 18479367, 47260890, respectively. Does lim_{k->inf} r(k) equal 2? - _Jon E. Schoenfield_, Mar 27 2018

%e The table below shows k, prime(k), the sum and average of the first k primes, and r(k), for each number k in the sequence, and also for k = 100, 1000, ..., 10^7.

%e .

%e n| a(n)=k prime(k) sum avg r(k)

%e --+--------------------------------------------------------

%e 1| 1 2 2 2.000 1.00000

%e 2| 2 3 5 2.500 1.20000

%e 3| 3 5 10 3.333 1.50000

%e 4| 4 7 17 4.250 1.64706

%e 5| 5 11 28 5.600 1.96429

%e 6| 7 17 58 8.286 2.05172

%e 7| 9 23 100 11.111 2.07000

%e 8| 10 29 129 12.900 2.24806

%e 9| 12 37 197 16.417 2.25381

%e 10| 17 59 440 25.882 2.27955

%e 11| 25 97 1060 42.400 2.28774

%e 12| 31 127 1720 55.484 2.28895

%e 13| 35 149 2276 65.029 2.29130

%e 14| 48 223 4661 97.104 2.29650

%e 100 541 24133 241.330 2.24174

%e 1000 7919 3682913 3682.913 2.15020

%e 10000 104729 496165411 49616.541 2.11077

%e 100000 1299709 62260698721 622606.987 2.08753

%e 1000000 15485863 7472966967499 7472966.967 2.07225

%e 10000000 179424673 870530414842019 87053041.484 2.06110

%Y Cf. A000040 (primes), A007504 (sum of first n primes), A006988 ((10^n)-th prime), A099824 (sum of first 10^n primes).

%K nonn,fini,full

%O 1,2

%A _Jon E. Schoenfield_, Mar 24 2018