cp's OEIS Frontend

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

%I A114829 #19 Feb 16 2025 08:33:00
%S A114829 1,2,3,4,6,8,11,14,18,23,29,36,44,53,63,74,87,101,117,135,155,177,201,
%T A114829 227,256,287,321,358,398,442,489,540,595,654,717,785,858,936,1019,
%U A114829 1107,1201,1301,1408,1521,1641,1768,1903,2046,2197,2356,2524,2701,2888,3085,3292,3510,3739,3979,4231
%N A114829 Each term is previous term plus floor of geometric mean of all previous terms.
%C A114829 What is this sequence, asymptotically?
%H A114829 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeometricMean.html">Geometric Mean.</a>
%F A114829 a(1) = 1, a(n+1) = a(n) + floor(GeometricMean[a(1),a(2),...,a(n)]).
%F A114829 a(n+1) = a(n) + floor((Product_{k=1..n} a(k))^(1/n)).
%e A114829 a(2) = 1 + floor(1^(1/1)) = 1 + 1 = 2.
%e A114829 a(3) = 2 + floor[(1*2)^(1/2)] = 2 + floor[sqrt(2)] = 2 + 1 = 3.
%e A114829 a(4) = 3 + floor[(1*2*3)^(1/3)] = 3 + floor[CubeRoot(6)] = 3 + 1 = 4.
%e A114829 a(5) = 4 + floor[(1*2*3*4)^(1/4)] = 4 + floor[4thRoot(24)] = 4 + 2 = 6.
%e A114829 a(6) = 6 + floor[(1*2*3*4*6)^(1/5)] = 6 + floor[5thRoot(144)] = 6 + 2 = 8.
%e A114829 a(7) = 8 + floor[(1*2*3*4*6*8)^(1/6)] = 6 + floor[6thRoot(1152)] = 8 + 3 = 11.
%p A114829 A114829 := proc(n)
%p A114829     option remember;
%p A114829     if n= 1 then
%p A114829         1;
%p A114829     else
%p A114829         mul(procname(i),i=1..n-1) ;
%p A114829         procname(n-1)+floor(root[n-1](%)) ;
%p A114829     end if;
%p A114829 end proc:
%p A114829 seq(A114829(n),n=1..60) ; # _R. J. Mathar_, Jun 23 2014
%t A114829 s={1};Do[AppendTo[s,Last[s]+Floor[GeometricMean[s]]],{n,58}];s (* _James C. McMahon_, Aug 19 2024 *)
%Y A114829 Cf. A065094, A065095.
%K A114829 easy,nonn
%O A114829 1,2
%A A114829 _Jonathan Vos Post_, Feb 19 2006