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

1, 2, 4, 7, 13, 23, 40, 70, 121, 210, 364, 631, 1093, 1894, 3281, 5683, 9844, 17050, 29532, 51151, 88597, 153455, 265792, 460366, 797377, 1381098, 2392132, 4143295, 7176398, 12429886, 21529195, 37289660, 64587586, 111868981, 193762759

Offset: 1

Author

Jonathan Vos Post, Feb 19 2006

Keywords

• easy
• nonn

Comments

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?

Examples

			a(3) = 2 + ceiling(2*1*2/(1+2)) = 2 + ceiling(4/3) = 2 + 2 = 4.
a(4) = 4 + ceiling(2*2*4/(2+4)) = 4 + ceiling(16/6) = 4 + 3 = 7.
a(5) = 7 + ceiling(2*4*7/(4+7)) = 7 + ceiling(56/8) = 7 + 6 = 13.
a(6) = 13 + ceiling(2*7*13/(7+13)) = 13 + ceiling(182/13) = 13 + 10 = 23.
a(7) = 23 + ceiling(2*13*23/(13+23)) = 23 + ceiling(598/36) = 23 + 17 = 40.
a(8) = 40 + ceiling(2*23*40/(23+40)) = 40 + ceiling(1840/63) = 40 + 30 = 70.
a(9) = 70 + ceiling(2*40*70/(40+70)) = 70 + ceiling(5600/110) = 70 + 51 = 121.
a(10) = 121 + ceiling(2*70*121/(70+121)) = 121 + ceiling(16940/191) = 121 + 89 = 210.
a(11) = 210 + ceiling(2*121*210/(121+210)) = 121 + ceiling(50820/331) = 210 + 154 = 364.
a(12) = 364 + ceiling(2*210*364/(210+364)) = 364 + ceiling(152880/574) = 364 + 267 = 631.
		

Links

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

Crossrefs

Cf. A065094, A065095.

Programs

Maple

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

Formula

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))).

Extensions

More terms from Emeric Deutsch, Mar 03 2006