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A062049 Integer part of geometric mean of first n primes. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Harry J. Smith, Table of n, a(n) for n=1..1000

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.

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A002110, A060620, A233824.

Sequence in context: A126027 A111581 A116572 * A025764 A237119 A011881

Adjacent sequences:  A062046 A062047 A062048 * A062050 A062051 A062052

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Jun 06 2001

EXTENSIONS

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

STATUS

approved

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Last modified October 1 17:53 EDT 2016. Contains 276659 sequences.