cp's OEIS Frontend

A114832 Each term is previous term plus ceiling of harmonic mean of two previous terms.

%I A114832 #9 Feb 16 2025 08:33:00
%S A114832 1,2,4,7,13,23,40,70,121,210,364,631,1093,1894,3281,5683,9844,17050,
%T A114832 29532,51151,88597,153455,265792,460366,797377,1381098,2392132,
%U A114832 4143295,7176398,12429886,21529195,37289660,64587586,111868981,193762759
%N A114832 Each term is previous term plus ceiling of harmonic mean of two previous terms.
%C A114832 For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y]. What is this sequence, asymptotically? a(n) is prime for n = 2, 4, 5, 6, 12, ... are there an infinite number of prime values?
%H A114832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean.</a>
%H A114832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeometricMean.html">Geometric Mean.</a>
%F A114832 a(1) = 1, a(2) = 2, for n > 2: a(n+1) = a(n) + ceiling(HarmonicMean(a(n), a(n-1))). a(n+1) = a(n) + ceiling((2*a(n)*a(n-1))/(a(n) + a(n-1))).
%e A114832 a(3) = 2 + ceiling(2*1*2/(1+2)) = 2 + ceiling(4/3) = 2 + 2 = 4.
%e A114832 a(4) = 4 + ceiling(2*2*4/(2+4)) = 4 + ceiling(16/6) = 4 + 3 = 7.
%e A114832 a(5) = 7 + ceiling(2*4*7/(4+7)) = 7 + ceiling(56/8) = 7 + 6 = 13.
%e A114832 a(6) = 13 + ceiling(2*7*13/(7+13)) = 13 + ceiling(182/13) = 13 + 10 = 23.
%e A114832 a(7) = 23 + ceiling(2*13*23/(13+23)) = 23 + ceiling(598/36) = 23 + 17 = 40.
%e A114832 a(8) = 40 + ceiling(2*23*40/(23+40)) = 40 + ceiling(1840/63) = 40 + 30 = 70.
%e A114832 a(9) = 70 + ceiling(2*40*70/(40+70)) = 70 + ceiling(5600/110) = 70 + 51 = 121.
%e A114832 a(10) = 121 + ceiling(2*70*121/(70+121)) = 121 + ceiling(16940/191) = 121 + 89 = 210.
%e A114832 a(11) = 210 + ceiling(2*121*210/(121+210)) = 121 + ceiling(50820/331) = 210 + 154 = 364.
%e A114832 a(12) = 364 + ceiling(2*210*364/(210+364)) = 364 + ceiling(152880/574) = 364 + 267 = 631.
%p A114832 a[1]:=1: a[2]:=2: for n from 2 to 40 do a[n+1]:=a[n]+ceil((2*a[n]*a[n-1])/(a[n]+a[n-1])) od: seq(a[n],n=1..40); # _Emeric Deutsch_, Mar 03 2006
%Y A114832 Cf. A065094, A065095.
%K A114832 easy,nonn
%O A114832 1,2
%A A114832 _Jonathan Vos Post_, Feb 19 2006
%E A114832 More terms from _Emeric Deutsch_, Mar 03 2006