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Arises from critical number of finite Abelian groups.
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%I #25 Oct 03 2024 07:57:02

%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.

%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