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 A145826 Arises from critical number of finite Abelian groups. 1
 7, 8, 11, 14, 19, 21, 26, 29, 34, 41, 43, 50, 55, 57, 62, 68, 75, 77, 84, 89, 91, 98, 102, 109, 117, 122, 124, 128, 131, 135, 150, 155, 161, 163, 174, 176, 183, 189, 194, 200, 206, 209, 219, 221, 226, 228, 241, 254, 258, 260, 264, 271, 273, 283, 290, 296, 302 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 P. Erdős 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, arXiv:0810.3223 [math.NT], Oct 17, 2008. FORMULA a(n) = prime(n) + floor(2*(sqrt(prime(n)+2))) + 1, where prime(n) = n-th prime = A000040(n). a(n)>= A000006(n) + A008864(n). [R. J. Mathar, Jan 05 2009] EXAMPLE 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. MAPLE map(t -> t + floor(2*sqrt(t+2))+1, [seq(ithprime(i), i=1..100)]); # Robert Israel, Feb 02 2016 MATHEMATICA Table[Prime[n] + Floor[2 (Sqrt[Prime[n] + 2])] + 1, {n, 60}] (* Vincenzo Librandi, Feb 02 2016 *) PROG (PARI) a(n) = prime(n) + floor(2*(sqrt(prime(n)+2))) + 1; \\ Michel Marcus, Feb 01 2016 (Magma) [NthPrime(n)+Floor(2*(Sqrt(NthPrime(n)+2)))+1: n in [1..80]]; // Vincenzo Librandi, Feb 02 2016 CROSSREFS Cf. A000040. Sequence in context: A309592 A090385 A350694 * A102963 A117619 A226977 Adjacent sequences: A145823 A145824 A145825 * A145827 A145828 A145829 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Oct 20 2008 EXTENSIONS More terms from R. J. Mathar, Jan 05 2009 STATUS approved

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Last modified August 4 01:40 EDT 2024. Contains 374905 sequences. (Running on oeis4.)