A174682 Positive integers which cannot be represented as half-sums (averages) of two primes with prime subscripts.

1, 2, 6, 9, 12, 13, 15, 16, 19, 20, 25, 27, 28, 30, 33, 34, 37, 40, 46, 48, 51, 52, 53, 55, 58, 61, 64, 68, 73, 74, 76, 77, 78, 82, 85, 86, 89, 90, 100, 102, 103, 106, 113, 115, 117, 124, 128, 130, 132, 134, 138, 145, 146, 148, 149, 151, 152, 155, 156, 158, 161, 163, 164

Offset: 1

Author

Jonathan Vos Post, Mar 26 2010

Keywords

• easy
• nonn

Comments

Jason Kimberley computed the first 733 positive integers which cannot be represented as the half sum of two primes with prime subscripts (A174682) as found using the first 998 values of A006450. From computing the first 20 thousand terms of A006450 (the 20000th term is 3118459), he shows the next value in the sequence of complements must be greater than 2907940. The PIP-Goldbach Conjecture is: every sufficiently large even number can be represented as the sum of two primes with prime subscripts.

Examples

			a(1) = 1 and a(2) = 2 are in the sequence because they are smaller than the first half-sum (average) of two primes with prime subscripts 3 = (3 + 3)/2 because 3 is the first prime with prime subscript, p(2). a(3) = 6 because there is no such half-sum between (5 + 5)/2 = 5 and (3 + 11)/2 = 7.
		

Links

• Jason Kimberley, Table of n, a(n) for n = 1..733

Crossrefs

Cf. A000040, A006450, A174681.

Formula

Complement of A174681. Complement of {(A006450(i) + A006450(j))/2} = Complement of {(A000040(A000040(i)) + A000040(A000040(j)))/2}.