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%I #30 Dec 11 2020 23:17:53
%S 0,1,0,0,1,0,1,0,1,1,1,0,2,0,1,1,2,0,3,0,1,1,3,0,3,1,1,1,5,0,6,0,2,2,
%T 3,1,7,0,3,1,8,0,9,1,1,2,9,0,8,1,3,2,11,0,7,1,4,3,13,0,14,1,3,4,8,1,
%U 15,1,6,1,16,0,17,3,2,4,11,1,18,0,7,4,19,0
%N a(n) = number of ways that one can write n as the sum of two positive integers such that i) the integers are relatively prime to n but ii) the integers are not themselves prime.
%C These might be called "relative Goldbach partitions."
%C This sequence was first discovered by my student Houston Hutchinson.
%C We became interested in this sequence when looking at Goldbach Partitions thus at first we only considered the even numbered terms. The graph of the even values of a(n) looks like Goldbach's comet except with an exponential appearance rather than a logarithmic appearance. We give a formula for the even values in the formula section.
%C Sequence A141095 has the terms for even n.
%H T. D. Noe, <a href="/A185279/b185279.txt">Table of n, a(n) for n = 1..5000</a>
%F For even n >= 4, denote the number of Goldbach partitions that have distinct primes by g(n), denote the totient of n by t(n), and denote the primes less than n that are NOT factors of n by p(n). Then a(n) = g(n)- p(n) + t(n)/2.
%F a(n) = Sum_{i=1..floor(n/2)} [GCD(i, n-i) = 1] * c(i) * c(n-i), where c is the characteristic function of nonprimes (A005171) and [ ] is the Iverson bracket. - _Wesley Ivan Hurt_, Dec 08 2020
%e a(34) is the first even term with value greater than 1. The number 34 = 33 + 1 and 25 + 9. The latter sums meet the requirements listed in the definition. For odd n greater than 3, a(n) will always be at least 1 since 1 + (n - 1) is a sum that satisfies the definition. For example a(5) = 1 since 5 = 1 + 4.
%t Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 100}] (* _T. D. Noe_, Dec 05 2013 *)
%o (Sage)
%o def A185279(n):
%o return sum(1 for i in (1..n//2) if all(gcd(j,n) == 1 and not is_prime(j) for j in (i, n-i))) # _D. S. McNeil_, Mar 05 2011
%Y Cf. A005171, A141095.
%K nonn
%O 1,13
%A _Jason Holland_, Feb 19 2011
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