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

Author

Jonathan Vos Post, Feb 19 2006

Keywords

• easy
• nonn

Comments

What is this sequence, asymptotically?

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Geometric Mean.

Crossrefs

Cf. A065094, A065095.

Programs

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

Mathematica

Nest[Append[#,Last[#]+Ceiling@GeometricMean[#]]&,{1},58] (* James C. McMahon, Aug 20 2024 *)

Formula

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

a(n+1) = a(n) + ceiling((Product_{k=1..n} a(k))^(1/n)).