1,1

Freeze, Gao, Geroldinger abstract: Let G be an additive, finite Abelian group. The critical number cr(G) of G is the smallest positive integer L such that for every subset S of {G setminus 0} with |S| => L the following holds: Every element of G can be written as a nonempty sum of distinct elements from S. The critical number was first studied by P. Erdos and H. Heilbronn in 1964 and due to the contributions of many authors the value of cr(G) is known for all finite Abelian groups G except for G == Z}/pq{Z where p, q are primes such that p+floor(2 sqrt{p-2})+1 < q < 2p. We determine that cr(G) = p+q-2 for such groups.

P. Erdos and H. Heilbronn, On the addition of residue classes modulo p, Acta Arith. 9 (1964), 149 - 159.

Michael Freeze, Weidong Gao and Alfred Geroldinger, The critical number of finite Abelian groups, Oct 17, 2008.

a(n) = p(n) + floor(2*(sqrt(p(n)+2)) + 1, where p(n) = n-th prime = A000040(n).

a(n)>= A000006(n)+A008864(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 05 2009]

a(10) = p(10) + floor(2*(sqrt(p(10)+2)) + 1 = 29 + floor(2*(sqrt(29+2)) + 1 = 29 + floor(2*5.56776436) + 1 = 29 + floor(11.1355287) + 1 = 29 + 11 + 1 = 41.

Cf. A000040.

Sequence in context: A071117 A054221 A090385 this_sequence A102963 A117619 A098731

Adjacent sequences: A145823 A145824 A145825 this_sequence A145827 A145828 A145829

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 20 2008

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 05 2009