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A364879
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a(n) is the smallest number k such that (sum of composites <= k) / (sum of primes <= k) >= n.
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1
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2, 6, 10, 28, 126, 520, 1394, 4440, 11765, 35702, 98202, 271718, 736814, 2012631, 5478367, 14867499, 40448112, 109944053, 298170203, 810416222, 2200884471, 5980529528
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = min {k : (Sum_{c<=k, c composite} c)/(Sum_{p<=k, p prime} p) >= n}.
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EXAMPLE
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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:
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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
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PROG
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(Python)
from itertools import count
from sympy import isprime
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:
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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