A364879 a(n) is the smallest number k such that (sum of composites <= k) / (sum of primes <= k) >= n.

2, 6, 10, 28, 126, 520, 1394, 4440, 11765, 35702, 98202, 271718, 736814, 2012631, 5478367, 14867499, 40448112, 109944053, 298170203, 810416222, 2200884471, 5980529528

Offset: 0

Author

Jon E. Schoenfield, Sep 10 2023

Keywords

• nonn
• more

Comments

a(n)+1 is a prime for n = 0, 1, 2, 3, 4, 5, and 7 (thus, for n = 1, 2, 3, 4, 5, and 7, a(n) is the last of a run of consecutive composites), but not for n = 6, nor for any n in 8..16.

For n > 0, a(n) is at least the n-th in a run of consecutive composites. a(15) is the 58th in a run of 71 consecutive composites.

Examples

			Let Sp(k) and Sc(k) be the sums of the primes <= k and the composites <= k, respectively. Then the sums and ratios begin as follows:
.
   k | Sp(k) | Sc(k) | Sc(k)/Sp(k)
  ---+-------+-------+------------
   1 |     0 |     0 | (undefined)
   2 |     2 |     0 |  0/2  = 0         so a(0) =  2
   3 |     5 |     0 |  0/5  = 0
   4 |     5 |     4 |  4/5  = 0.8
   5 |    10 |     4 |  4/10 = 0.4
   6 |    10 |    10 | 10/10 = 1         so a(1) =  6
   7 |    17 |    10 | 10/17 = 0.5882...
   8 |    17 |    18 | 18/17 = 1.0588...
   9 |    17 |    27 | 27/17 = 1.5882...
  10 |    17 |    37 | 37/17 = 2.1764... so a(2) = 10
		

Crossrefs

Cf. A000040, A002808, A007504, A034387, A053767, A101256.

Programs

Python

from itertools import count
from sympy import isprime
def A364879(n):
    c, cn, m = 0, 0, n+1<<1
    for k in count(2):
        if isprime(k):
            c += k
            cn += k*m
        if k*(k+1)-1 >= cn:
            return k # Chai Wah Wu, Sep 10 2023

Formula

a(n) = min {k : (Sum_{c<=k, c composite} c)/(Sum_{p<=k, p prime} p) >= n}.

a(n) = min {k>1 : k(k+1)-1>=2*A034387(k)*(n+1)}. - Chai Wah Wu, Sep 10 2023

Extensions

a(17)-a(21) from Chai Wah Wu, Sep 10 2023