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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 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.)