OFFSET
1,2
COMMENTS
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.
a(14)=48 is the final term. Beyond k=48, r(k) decreases fairly smoothly (although nonmonotonically); see the Example section.
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
EXAMPLE
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.
.
n| a(n)=k prime(k) sum avg r(k)
--+--------------------------------------------------------
1| 1 2 2 2.000 1.00000
2| 2 3 5 2.500 1.20000
3| 3 5 10 3.333 1.50000
4| 4 7 17 4.250 1.64706
5| 5 11 28 5.600 1.96429
6| 7 17 58 8.286 2.05172
7| 9 23 100 11.111 2.07000
8| 10 29 129 12.900 2.24806
9| 12 37 197 16.417 2.25381
10| 17 59 440 25.882 2.27955
11| 25 97 1060 42.400 2.28774
12| 31 127 1720 55.484 2.28895
13| 35 149 2276 65.029 2.29130
14| 48 223 4661 97.104 2.29650
100 541 24133 241.330 2.24174
1000 7919 3682913 3682.913 2.15020
10000 104729 496165411 49616.541 2.11077
100000 1299709 62260698721 622606.987 2.08753
1000000 15485863 7472966967499 7472966.967 2.07225
10000000 179424673 870530414842019 87053041.484 2.06110
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Jon E. Schoenfield, Mar 24 2018
STATUS
approved