cp's OEIS Frontend

A062049 Integer part of geometric mean of first n primes.

%I A062049 #45 Jul 19 2017 03:08:39
%S A062049 2,2,3,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,20,21,23,24,25,27,28,29,
%T A062049 31,32,34,35,36,38,39,41,43,44,46,47,49,50,52,53,55,57,58,60,61,63,65,
%U A062049 66,68,70,71,73,75,76,78,80,82,83,85,87,88,90,92,94,95,97,99,101,103
%N A062049 Integer part of geometric mean of first n primes.
%C A062049 For large n, the ratio prime(n)/a(n) tends to e (very slowly). This was conjectured by Anton Vrba in 2010 (see Rivera, 2010) and proved by Sandor and Verroken (2011). Tighter bounds and asymptotics for a(n) are proved in the note "On the geometric mean of the first n primes" (2016) (see links). Better formulas prime(n)/a(n) ~ exp(1 + several terms A233824(k) / log^k(prime(n))) exist for larger n; see examples in the formula section. - _Alexei Kourbatov_, Feb 27 2016.
%H A062049 Harry J. Smith, <a href="/A062049/b062049.txt">Table of n, a(n) for n=1..1000</a>
%H A062049 Christian Axler, <a href="https://arxiv.org/abs/1609.07934">On the arithmetic and geometric means of the prime numbers</a>, arXiv:1609.07934 [math.NT], 2016.
%H A062049 A. Kourbatov, <a href="http://arxiv.org/abs/1603.00855">On the geometric mean of the first n primes</a>, arXiv:1603.00855 [math.NT], 2016.
%H A062049 C. Rivera, ed. <a href="http://www.primepuzzles.net/conjectures/conj_067.htm">Conjecture 67. Primes and e</a>, 2010.
%H A062049 J. Sandor and A. Verroken, <a href="http://nntdm.net/papers/nntdm-17/NNTDM-17-2-01-03.pdf">On a limit involving the product of prime numbers</a>, Notes Number Theory Discrete Math. 17 (2011), No. 2, 1-3.
%F A062049 From _Alexei Kourbatov_, Feb 22 2016: (Start)
%F A062049 a(n) ~ prime(n)/exp(1 + 1/log(prime(n)) + O(1/log^2(prime(n)))).
%F A062049 a(n) ~ prime(n)/e (this approximation is poor).
%F A062049 a(n) ~ prime(n)/exp(1 + 1/log(prime(n))).
%F A062049 a(n) ~ prime(n)/exp(1 + 1/log(prime(n)) + 3/log^2(prime(n))).
%F A062049 a(n) ~ prime(n)/exp(1 + 1/log(prime(n)) + 3/log^2(prime(n)) + 13/log^3(prime(n))).
%F A062049 a(n) < (1/2)*prime(n) for n>3.
%F A062049 (End)
%F A062049 a(n) = floor(A002110(n)^(1/n)). - _Michel Marcus_, Feb 22 2016
%e A062049 a(5) = floor( (2*3*5*7*11)^(1/5) ) = 4.
%p A062049 P:= 1:
%p A062049 A[0]:= 1:
%p A062049 for n from 1 to 100 do
%p A062049   P:= ithprime(n)*P;
%p A062049   for k from A[n-1] while (k+1)^n <= P do od:
%p A062049   A[n]:= k;
%p A062049 od:
%p A062049 seq(A[i],i=1..100); # _Robert Israel_, Feb 22 2016
%t A062049 With[{pl=Prime[Range[80]]},Table[IntegerPart[GeometricMean[Take[pl,n]]],{n,80}]] (* _Harvey P. Dale_, Mar 31 2012 *)
%o A062049 (PARI) { default(realprecision, 100); p=1; for (n=1, 1000, p*=prime(n); write("b062049.txt", n, " ", p^(1/n)\1) ) } \\ _Harry J. Smith_, Jul 30 2009
%Y A062049 Cf. A002110, A060620, A233824.
%K A062049 nonn
%O A062049 1,1
%A A062049 _Amarnath Murthy_, Jun 06 2001
%E A062049 More terms from Larry Reeves (larryr(AT)acm.org) and _Matthew Conroy_, Jun 11 2001