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%I #19 Sep 08 2022 08:45:38
%S 7,8,11,14,19,21,26,29,34,41,43,50,55,57,62,68,75,77,84,89,91,98,102,
%T 109,117,122,124,128,131,135,150,155,161,163,174,176,183,189,194,200,
%U 206,209,219,221,226,228,241,254,258,260,264,271,273,283,290,296,302
%N Arises from critical number of finite Abelian groups.
%C 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.
%H Robert Israel, <a href="/A145826/b145826.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Erdős and H. Heilbronn, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa9/aa9115.pdf">On the addition of residue classes modulo p</a>, Acta Arith. 9 (1964), 149 - 159.
%H Michael Freeze, Weidong Gao and Alfred Geroldinger, <a href="http://arxiv.org/abs/0810.3223">The critical number of finite Abelian groups</a>, arXiv:0810.3223 [math.NT], Oct 17, 2008.
%F a(n) = prime(n) + floor(2*(sqrt(prime(n)+2))) + 1, where prime(n) = n-th prime = A000040(n).
%F a(n)>= A000006(n) + A008864(n). [_R. J. Mathar_, Jan 05 2009]
%e a(10) = prime(10) + floor(2*(sqrt(prime(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.
%p map(t -> t + floor(2*sqrt(t+2))+1, [seq(ithprime(i),i=1..100)]); # _Robert Israel_, Feb 02 2016
%t Table[Prime[n] + Floor[2 (Sqrt[Prime[n] + 2])] + 1, {n, 60}] (* _Vincenzo Librandi_, Feb 02 2016 *)
%o (PARI) a(n) = prime(n) + floor(2*(sqrt(prime(n)+2))) + 1; \\ _Michel Marcus_, Feb 01 2016
%o (Magma) [NthPrime(n)+Floor(2*(Sqrt(NthPrime(n)+2)))+1: n in [1..80]]; // _Vincenzo Librandi_, Feb 02 2016
%Y Cf. A000040.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Oct 20 2008
%E More terms from _R. J. Mathar_, Jan 05 2009
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