A185279 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.

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, 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, 15, 1, 6, 1, 16, 0, 17, 3, 2, 4, 11, 1, 18, 0, 7, 4, 19, 0

Offset: 1

Author

Jason Holland, Feb 19 2011

Keywords

• nonn

Comments

These might be called "relative Goldbach partitions."

This sequence was first discovered by my student Houston Hutchinson.

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.

Sequence A141095 has the terms for even n.

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A005171, A141095.

Programs

Mathematica

Table[Length[Select[Range[n/2], ! PrimeQ[#] && ! PrimeQ[n - #] && GCD[#, n - #] == 1 &]], {n, 100}] (* T. D. Noe, Dec 05 2013 *)

Sage

def A185279(n):
    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

Formula

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.

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