A062049 Integer part of geometric mean of first n primes.

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, 31, 32, 34, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 75, 76, 78, 80, 82, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 103

Offset: 1

Author

Amarnath Murthy, Jun 06 2001

Keywords

• nonn

Comments

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.

Examples

			a(5) = floor( (2*3*5*7*11)^(1/5) ) = 4.
		

Links

• Harry J. Smith, Table of n, a(n) for n=1..1000
• Christian Axler, On the arithmetic and geometric means of the prime numbers, arXiv:1609.07934 [math.NT], 2016.
• A. Kourbatov, On the geometric mean of the first n primes, arXiv:1603.00855 [math.NT], 2016.
• C. Rivera, ed. Conjecture 67. Primes and e, 2010.
• J. Sandor and A. Verroken, On a limit involving the product of prime numbers, Notes Number Theory Discrete Math. 17 (2011), No. 2, 1-3.

Crossrefs

Cf. A002110, A060620, A233824.

Programs

Maple

P:= 1:
A[0]:= 1:
for n from 1 to 100 do
  P:= ithprime(n)*P;
  for k from A[n-1] while (k+1)^n <= P do od:
  A[n]:= k;
od:
seq(A[i],i=1..100); # Robert Israel, Feb 22 2016

Mathematica

With[{pl=Prime[Range[80]]},Table[IntegerPart[GeometricMean[Take[pl,n]]],{n,80}]] (* Harvey P. Dale, Mar 31 2012 *)

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

Formula

From Alexei Kourbatov, Feb 22 2016: (Start)

a(n) ~ prime(n)/exp(1 + 1/log(prime(n)) + O(1/log^2(prime(n)))).

a(n) ~ prime(n)/e (this approximation is poor).

a(n) ~ prime(n)/exp(1 + 1/log(prime(n))).

a(n) ~ prime(n)/exp(1 + 1/log(prime(n)) + 3/log^2(prime(n))).

a(n) ~ prime(n)/exp(1 + 1/log(prime(n)) + 3/log^2(prime(n)) + 13/log^3(prime(n))).

a(n) < (1/2)*prime(n) for n>3.

(End)

a(n) = floor(A002110(n)^(1/n)). - Michel Marcus, Feb 22 2016

Extensions

More terms from Larry Reeves (larryr(AT)acm.org) and Matthew Conroy, Jun 11 2001