A119494 - OEIS

%I #23 Dec 24 2018 06:11:38

%S 5,29,109,347,857,1627,2999,4931,7759,11677,16111,22229,29269,37717,

%T 48527,61057,75503,91463,110567,131671,155509,183587,214189,248597,

%U 286073,325889,369983,419459,473659,534043,600631,667547,739549,816779,901007,988661

%N a(n) = smallest prime number p_k such that 1/p_n + 1/p_{n+1} + ... + 1/p_k > 1.

%C Domaratzki, Ellul, Shallit, & Wang call the n-th term of A092325 ϖ(n), and A092325(n) = pi(a(n)). - _Charles R Greathouse IV_, Aug 08 2016

%D J.-M. De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 76, entry 347 and page 108, entry 857.

%H Amiram Eldar, <a href="/A119494/b119494.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Domaratzki, Keith Ellul, Jeffrey Shallit and Ming-Wei Wang, <a href="http://dx.doi.org/10.1142/S0129054105003352">Non-Uniqueness and Radius of Cyclic Unary NFAs</a>, International Journal of Foundations of Computer Science, Vol. 16, No. 5 (2005) pp. 883-896, <a href="http://www.cs.umanitoba.ca/~mdomarat/pubs/DESW_dcfs.pdf">alternative link</a>.

%F a(n) is approximately prime(n)^e.

%F a(n) = prime(A092325(n)). - _Amiram Eldar_, Dec 24 2018

%e a(2) = 29 because 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 + 1/29 = 1.0334... > 1 and 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 = 0.9989... < 1.

%t f[0]={0,0}; f[n_] := f[n] = Module[{f1=f[n-1]}, p=f1[[1]]; s=f1[[2]]-If[n>1, 1/Prime[n-1], 0]; While[s<1, p=NextPrime[p]; s+=1/p]; {p,s}]; f[#][[1]] & /@ Range[30] (* _Amiram Eldar_, Dec 24 2018 *)

%o (PARI) a(n)=my(s=0.);forprime(p=prime(n),default(primelimit),s+=1/p;if(s>1,return(p)))

%Y Cf. A092325, A016088.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, May 25 2006

%E Definition corrected by _Ray Chandler_, Jun 09 2006

%E Edited by _Charles R Greathouse IV_, Nov 12 2009

%E a(35)-a(36) from _Amiram Eldar_, Dec 24 2018