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A114830
Each term is previous term plus ceiling of geometric mean of all previous terms.
1
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?
LINKS
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((Product_{k=1..n} a(k))^(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
MATHEMATICA
Nest[Append[#, Last[#]+Ceiling@GeometricMean[#]]&, {1}, 58] (* James C. McMahon, Aug 20 2024 *)
CROSSREFS
Sequence in context: A247179 A319158 A175780 * A177239 A001304 A000064
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 19 2006
STATUS
approved