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A301599 Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high. 0
1, 2, 3, 4, 5, 7, 9, 10, 12, 17, 25, 31, 35, 48 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..14.

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

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

Sequence in context: A320848 A228234 A060526 * A036408 A274583 A055600

Adjacent sequences:  A301596 A301597 A301598 * A301600 A301601 A301602

KEYWORD

nonn,fini,full

AUTHOR

Jon E. Schoenfield, Mar 24 2018

STATUS

approved

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Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)