A114830 Each term is previous term plus ceiling of geometric mean of all previous terms.

1, 2, 4, 6, 9, 13, 18, 24, 31, 39, 48, 59, 71, 85, 101, 119, 139, 162, 187, 215, 246, 280, 318, 359, 404, 453, 507, 565, 628, 697, 771, 851, 937, 1029, 1128, 1234, 1348, 1470, 1600, 1738, 1885, 2042, 2209, 2386, 2574, 2773, 2984, 3207, 3443, 3692, 3955, 4232, 4524, 4831, 5154, 5494, 5851, 6226, 6620

Offset

1,2

Comments

What is this sequence, asymptotically? a(n) is prime for n = 2, 6, 9, 12, 13, 15, 17, 24, ... are there an infinite number of prime values?

Links

Table of n, a(n) for n=1..59.

Eric Weisstein's World of Mathematics, Geometric Mean.

Formula

a(1) = 1, a(n+1) = a(n) + ceiling(GeometricMean[a(1),a(2),...,a(n)]). a(n+1) = a(n) + ceiling[((a(1)*a(2)*,...,*a(n))^(1/n)].

Example

a(2) = 1 + ceiling(1^(1/1)) = 1 + 1 = 2.
a(3) = 2 + ceiling[(1*2)^(1/2)] = 2 + ceiling[sqrt(2)] = 2 + 2 = 4.
a(4) = 4 + ceiling[(1*2*4)^(1/3)] = 4 + ceiling[CubeRoot(8)] = 4 + 2 = 6.
a(5) = 6 + ceiling[(1*2*4*6)^(1/4)] = 6 + floor[4thRoot(48)] = 6 + 3 = 9.
a(6) = 9 + ceiling[(1*2*4*6*9)^(1/5)] = 9 + ceiling[5thRoot(432)] = 9 + 4 = 13.
a(7) = 13 + ceiling[(1*2*4*6*9*13)^(1/6)] = 6 + floor[6thRoot(5616)] = 13 + 5 = 18.
a(25) = 359 + ceiling[(1 * 2 * 4 * 6 * 9 * 13 * 18 * 24 * 31 * 39 * 48 * 59 * 71 * 85 * 101 * 119 * 139 * 162 * 187 * 215 * 246 * 280 * 318 * 359)^(1/24)] = 359 + ceiling[44.8074289] = 359 + 45 = 404.

Maple

A114830 := proc(n)
    option remember;
    if n= 1 then
        1;
    else
        mul(procname(i), i=1..n-1) ;
        procname(n-1)+ceil(root[n-1](%)) ;
    end if;
end proc:
seq(A114830(n), n=1..60) ; # R. J. Mathar, Jun 23 2014

Crossrefs

Cf. A065094, A065095.

Sequence in context: A247179 A319158 A175780 * A177239 A001304 A000064

Adjacent sequences: A114827 A114828 A114829 * A114831 A114832 A114833

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 19 2006

Status

approved