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

1,2

Comments

What is this sequence, asymptotically?

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

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

Example

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.

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 *)

Crossrefs

Cf. A065094, A065095.

Sequence in context: A238383 A134953 A175870 * A175869 A007279 A034891

Adjacent sequences: A114826 A114827 A114828 * A114830 A114831 A114832

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 19 2006

Status

approved