2,3

Suppose that A is a subset of (1,...,N) is such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^(1/4))) for some absolute c>0.

Imre Z. Ruzsa, Tom Sanders, Difference sets and the primes, October 2, 2007.

a(1) is undefined because we cannot have a difference between two elements of a set of 1 element.

a(2) = 0 because the only subset of size =>2 of (1,2) is (1,2), and 2-1 = 1 is 1 less thasn the prime 2.

a(3) = 0 because the only subset of size =>2 of (1,2,3) are (1,2), and 2-1 = 1 is 1 less thasn the prime 2; (1,3), and 3-1 = 2 is 1 less thasn the prime 3; and (2,3) and and 3-2 = 1 is 1 less thasn the prime 2.

a(4) = 2 because (1,4) is the unique subset of (1,2,3,4) with the desired property that 4-1 = 3 is not 1 less than a prime.

a(9) = 3 because (1,4,9) is the unique subset of (1,2,3,4,5,6,7,8,9) with the desired property that 4-1 = 3 is not 1 less than a prime, and 9-1 = 8 is not 1 less than a prime, and 9-4 = 5 is not 1 less than a prime.

For n=9, 10 and 11, the cardinality is limited to 3 (the subset {1,4,9}). For

12 <= n <= 17, the cardinality is limited to 4 (the subset {1,4,9,12}).

Cf. A000040.

Sequence in context: A005857 A025809 A114575 this_sequence A090735 A090736 A094999

Adjacent sequences: A131846 A131847 A131848 this_sequence A131850 A131851 A131852

more,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 04 2007

Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 15 2008