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%I #18 Sep 11 2023 01:47:14
%S 2,6,10,28,126,520,1394,4440,11765,35702,98202,271718,736814,2012631,
%T 5478367,14867499,40448112,109944053,298170203,810416222,2200884471,
%U 5980529528
%N a(n) is the smallest number k such that (sum of composites <= k) / (sum of primes <= k) >= n.
%C 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.
%C 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.
%F a(n) = min {k : (Sum_{c<=k, c composite} c)/(Sum_{p<=k, p prime} p) >= n}.
%F a(n) = min {k>1 : k(k+1)-1>=2*A034387(k)*(n+1)}. - _Chai Wah Wu_, Sep 10 2023
%e 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:
%e .
%e k | Sp(k) | Sc(k) | Sc(k)/Sp(k)
%e ---+-------+-------+------------
%e 1 | 0 | 0 | (undefined)
%e 2 | 2 | 0 | 0/2 = 0 so a(0) = 2
%e 3 | 5 | 0 | 0/5 = 0
%e 4 | 5 | 4 | 4/5 = 0.8
%e 5 | 10 | 4 | 4/10 = 0.4
%e 6 | 10 | 10 | 10/10 = 1 so a(1) = 6
%e 7 | 17 | 10 | 10/17 = 0.5882...
%e 8 | 17 | 18 | 18/17 = 1.0588...
%e 9 | 17 | 27 | 27/17 = 1.5882...
%e 10 | 17 | 37 | 37/17 = 2.1764... so a(2) = 10
%o (Python)
%o from itertools import count
%o from sympy import isprime
%o def A364879(n):
%o c, cn, m = 0, 0, n+1<<1
%o for k in count(2):
%o if isprime(k):
%o c += k
%o cn += k*m
%o if k*(k+1)-1 >= cn:
%o return k # _Chai Wah Wu_, Sep 10 2023
%Y Cf. A000040, A002808, A007504, A034387, A053767, A101256.
%K nonn,more
%O 0,1
%A _Jon E. Schoenfield_, Sep 10 2023
%E a(17)-a(21) from _Chai Wah Wu_, Sep 10 2023
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