A301599 Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

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

Offset: 1

Author

Jon E. Schoenfield, Mar 24 2018

Keywords

• nonn
• fini
• full

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

Examples

			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

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