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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 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. 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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