A114829 Each term is previous term plus floor of geometric mean of all previous terms.

1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 44, 53, 63, 74, 87, 101, 117, 135, 155, 177, 201, 227, 256, 287, 321, 358, 398, 442, 489, 540, 595, 654, 717, 785, 858, 936, 1019, 1107, 1201, 1301, 1408, 1521, 1641, 1768, 1903, 2046, 2197, 2356, 2524, 2701, 2888, 3085, 3292, 3510, 3739, 3979, 4231

Offset: 1

Author

Jonathan Vos Post, Feb 19 2006

Keywords

• easy
• nonn

Comments

What is this sequence, asymptotically?

Examples

			a(2) = 1 + floor(1^(1/1)) = 1 + 1 = 2.
a(3) = 2 + floor[(1*2)^(1/2)] = 2 + floor[sqrt(2)] = 2 + 1 = 3.
a(4) = 3 + floor[(1*2*3)^(1/3)] = 3 + floor[CubeRoot(6)] = 3 + 1 = 4.
a(5) = 4 + floor[(1*2*3*4)^(1/4)] = 4 + floor[4thRoot(24)] = 4 + 2 = 6.
a(6) = 6 + floor[(1*2*3*4*6)^(1/5)] = 6 + floor[5thRoot(144)] = 6 + 2 = 8.
a(7) = 8 + floor[(1*2*3*4*6*8)^(1/6)] = 6 + floor[6thRoot(1152)] = 8 + 3 = 11.
		

Links

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

Crossrefs

Cf. A065094, A065095.

Programs

Maple

A114829 := proc(n)
    option remember;
    if n= 1 then
        1;
    else
        mul(procname(i),i=1..n-1) ;
        procname(n-1)+floor(root[n-1](%)) ;
    end if;
end proc:
seq(A114829(n),n=1..60) ; # R. J. Mathar, Jun 23 2014

Mathematica

s={1};Do[AppendTo[s,Last[s]+Floor[GeometricMean[s]]],{n,58}];s (* James C. McMahon, Aug 19 2024 *)

Formula

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

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