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A114832
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Each term is previous term plus ceiling of harmonic mean of two previous terms.
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0
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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
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OFFSET
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1,2
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COMMENTS
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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?
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LINKS
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FORMULA
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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))).
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EXAMPLE
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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.
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MAPLE
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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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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