A145826 Arises from critical number of finite Abelian groups.

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

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